m  MEMOmANi 
^jGor^':e  Davidson 
1825«1911 


Professor   of  Geof^raphy 
U)  iversity  of  California 


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ELEMEN 


■^X^ 


Differential     and     Integral 


CALCULUS, 


By  a  New  Method,  Founded  on  the  True  System  of 

Sir    Isaac  Newton,  without  the  Use    of 

Infinitesimals  or  Limits. 


REVISED    EDITION. 


By    C.    p.    BUCKINGHAM, 

AUTHOR    OF   THE     PRIiVCIPLES     OF     ARITHMETIC  :      FORMERLY    ASSISTANT   PROFESSOR     OF 

NATURAL    PHILOSOPHY  IN   TKE    U.    S.    MILITARY    ACADEMY,    AND     PROFESSOR    OF 

MATHEMATICS   AND   NATURAL   PHILOSOPHY   IN    KENYON   COLLEGE,  OHIO. 


CHICAGO : 
S.    C.    GRIGGS    &    COMPANY. 

1880. 


Entered  according  to  act  of  Congress,  in  the  year  1875,  by 

S.  C.  GRIGGS  &  CO., 

in  the  office  of  the  Librarian  of  Congress  at  Washington,  District  of  Columbia. 

Entered  according  to  act  of  Congress,  in  the  year  1880,  by 

S.  C   GRIGGS  &  CO., 

in  the  office  of  the  Librarian  of  Congress  at  Washington,  District  of  Columbia. 


Donnelley,  Cassette  &  Loyd,  Printers,  Clark  &  Adams  Streets. 


PREFACE  TO  THE  FIRST  EDITION. 


"  The  student  of  mathematics,  on  passing  from  the  lower 
branches  of  the  science  to  the  infinitesimal  analysis,  finds 
himself  in  a  strange  and  almost  wholly  foreign  department 
of  thought.  He  has  not  risen,  by  easy  and  gradual  steps, 
from  a  lower  to  a  higi.^^r,  purer  and  more  beautiful  region  of 
scientific  truth.  On  the  contrary,  he  is  painfully  impressed 
with  the  conviction,  that  the  continuity  of  the  science  has 
been  broken,  and  its  unity  destroyed,  by  the  influx  of  prin- 
ciples which  are  as  unintelligible  as  they  are  novel  He 
finds  himself  surrounded  by  enigmas  and  obscurities,  which 
only  serve  to  perplex  his  understanding  and  darken  his  aspi- 
rations after  knowledge."* 

He  fiixis  himself  required  to  ignore  the  principles  and 
axioms  that  have  hitherto  guided  his  studies  and  sustained 
his  convictions,  and  to  receive  in  their  stead  a  set  of  notions 
that  are  utterly  repugnant  to  all  his  preconceived  ideas  of 
truth.  When  he  is  told  that  one  quantity  may  be  added  to 
another  without  increasing  it,  or  subtracted  from  another 
without  diminishing  it — that  one  quantity  may  be  infinitely 
small,  and  another  infinitely  smaller,  and   another  infinitely 

*Bledsoe — Philosophy  of  Mathematics. 

ivi513314 


IV-  PREFACE. 

smaller  still,  and  so  on  ad  inJimMin  —  \h^\.  a  quantity  may 
be  so  small  that  it  can  not  be  divided,  and  yet  may  contain 
another  an  indefinite,  and  even  an  infinite  number  of  times  — 
<  hat  zero  is  not  always  nothing,  but  may  not  only  be  some- 
thing or  nothing  as  occasion  may  require,  but  may  be  both  at 
the  same  time  in  the  same  equation— it  is  not  surprising  that 
he  should  become  bewildered  and  disheartened.  Neverthe- 
less, if  he  study  the  text  books  that  are  considered  orthodox 
in  this  country  and  in  Europe  he  will  find  some  of  these 
notions  set  forth  in  them  all ;  not.  indeed,  in  their  naked 
deformity,  as  they  are  here  stated,  but  softened  and  made  as 
palatable  as  possible  by  associating  them  with,  or  concealing 
them  beneath,  propositions  that  are  undoubtedly  true. 

It  is,  indeed,  strange  that  a  science  so  exact  in  its  results 
should  have  its  principles  interwoven  with  so  much  that  is 
false  and  absurd  in  theory ;  especially  as  all  these  absurdi- 
ties have  been  so  often  exposed,  and  charged  against  the 
claims  of  the  calculus  as  a  true  science.  It  can  be  accounted 
for  only  by  the  influence  of  the  great  names  that  first  adopted 
them,  and  the  indisposition  of  mathematicians  to  depart 
from  the  simple  ideas  of  the  ancients  in  reference  to  the 
attributes  of  quantity.  They  regard  it  as  merely  inert, 
either  fixed  in  value  or  subject  only  to  such  changes  as  may 
be  arbitrarily  imposed  upon  it.  But  when  they  attempt  to 
carry  this  conception  into  the  operations  of  the  calculus, 
and  to  account  for  the  results  by  some  theory  consistent  with 
this  idea  of  quantity  they  are  inevitably  entangled  in  some 
of  the  absurd  notions  that  have  been  mentioned.  Many 
efforts  have  indeed  been  made  to  escape  such  glaring  incon- 


PREFACE.  V 

sistencies,  but  they  have  only  resulted  in  a  partial  success 
in  concealing  them. 

To  clear  the  way  for  a  logical  and  rational  consideration 
of  the  subject,  we  must  begin  with  the  fundamental  idea  of 
the  conditions  under  which  quantity  may  exist.  We  must, 
for  the  purposes  of  the  calculus,  consider  it  not  only  as  ca- 
pable of  being  increased  or  diminished ,  but  also  as  being 
actually  in  a  state  of  change.  It  must  (so  to  speak)  be  vital- 
ized., so  that  it  shall  be  endowed  with  tendencies  to  change  its 
value,  and  the  rate  and  direction  of  these  tendencies  will 
be  found  to  constitute  the  ground  work  of  the  whole  system. 
The  differential  calculus  is  the  science  of  rates,  and  its 
peculiar  subject  is  quantity  in  a  state  of  change. 

It  is  an  error,  therefore,  to  suppose,  as  has  often  been 
said,  that  the  "  7-ediictio  ad  absurduni,''  or  "  method  of  ex- 
haustion," of  the  ancient  mathematicians,  contains  the 
germ  of  the  differential  calculus.  This  hallucination  has 
arisen  from  the  same  source  as  the  false  notions  before 
mentioned.  That  peculiar  attribute  of  quantity  upon  which 
the  transcendental  analysis  was  built,  never  found  a  2:)lace 
among  the  ideas  of  the  Greek  Philosophers  ;  and  even  Leib- 
nitz, the  competitor  of  Newton  for  the  honor  of  the  inven- 
tion, and  who  was  the  first  to  construct  a  system  of  rules 
for  the  analytical  machinery  of  the  science,  never  got  beyond 
the  ancient  conception  of  the  conditions  of  (juantity.  and, 
therefore,  gave,  says  Comte,  "  an  explanation  entirely  erro- 
neous "—he  never  comprehended  the  true  philosophical 
basis  of  his  own  system. 

The  only  original  birth-place  of  the  fundamental  idea  oi 


VI  PREFACE. 

quantity  which  forms  the  true  germ  of  the  calculus,  was  in 
the  mind  of  the  immortal  Newton.  Starting  with  this  idea, 
the  results  of  the  calculus  follow  logically  and  directly 
through  the  beaten  track  of  mathematical  thought,  with 
that  clearness  of  evidence  which  has  ever  been  the  boast  of 
mathematics,  and  which  leaves  neither  doubt  nor  distrust 
in  the  mind  of  the  student. 

To  develop  this  idea  is  the  object  of  this  work. 

C    P.  BUCKINGHAM. 

Chicago,  Jan.  i,  1875. 


PREFACE  TO  THE  PRESENT  EDITION. 


In  a  former  edition  of  this  work,  I  have  condemned 
both  the  method  of  infinitesimals  and  the  method  of  limits, 
as  the  true  exponents  of  the  Differential  Calculus  ;  the  first, 
because  its  principles  are  manifestly  false  ;  and  the  latter, 
because  no  satisfactory  demonstration  had  been  given,  that 
its  assumptions  were  mathematically   true. 

Subsequent  reflection,  while  it  has  not  changed  my  opin- 
ion as  to  the  weakness  of  the  demonstrations  or  arguments 
usually  advanced  in  support  of  the  method  of  limits,  has 
convinced  me  that  by  adopting  a  right  conception  of  the 
true  principles  of  the  calculus,  we  may  fathcjm  the  mystery 
of  that  method,  and  give  it  an  honored  place  in  the  temple 
of  scientific  truth. 

The  method  of  limits  has  usually  been  regarded  as  rest- 
ing upon  the  principle  of  exhaustion,  and  when  demon- 
strated or  defended  on  that  ground,  the  attempt  has  always 
been  a  failure.  It  therefore  seemed  to  me,  that  when  such 
men  as  Carnot,  Lagrange,  D'Alembert,  Maclaurin  and  even 
Newton  himself,  as  well  as  a  host  of  others,  after  a  discus- 
sion during  two  hundred  years,  had  failed  to  give  a  rational 

demonstration  of  the  true  meaning  of  -  =  2X,  as  it  arises  in 

o 

the  calculus,  it  must  be  impossible  to  do  it.  Hence  I  took 
that  ground,  and,  on  it,  objected  to  the  method  of  limits  as 
defective  in  its  logic,  and  hence  without  a  valid  claim  to  a 
place  in  mathematics. 

I  have  now,  however,  constructed  a  rigorous  mathemat- 
ical  demonstration,  entirely  free  from  all  metaphysical   rea- 


PREFACE. 


soning  and  which,  I  believe,  is   now   set  before   the   public, 

•       o 
/^r //^^^  /fr^v //;//t',  showiHiy  that  the  equation  —   =   2x    is    not 

o 

only    significant,  but   that  2x  in    that   equation   is   the    true 

differential    coefficient,  and   that  the    hitherto  unauthorized 

substitution  of  ^—  for— is   perfectly  justifiable.     This    de- 

dx  o 

monstration,  which  will  be  found  in  these  pages,  is  based  on 
the  principle  which  is  at  the  foundation  of  this  work. 

The  practical  value  of  this  method,  even  when  proved  to 
be  right  in  principle,  must  be  estimated  by  considerations 
which  will  be  set  forth  in-  another  place. 

C.  P.  Buckingham. 


Chicago,  July  i,  iSSo. 


Contents. 


INTRODUCTION. 

PAGE 

Early  State  o.  Geometry _ 3 

Method  of  Exhaustion 4 

Analytical  Geometry 6 

Method  of  I ndivisibles  -._ 8 

Method  of  Infinitesimals 14 

Method  of  Limits -  - 20 

Prime  and  Ultimate  Ratios 24 

Fluxions;  or  the  True  Method  of  Newton 35 


PART    I 


DIFFERENTIAL    CALCULUS. 

Section  I,     Definitions  and  First  Principles 49 

Variables  Defined 49 

Rate  of  Variation - — 50 

Differentials 51 

Constants 52 

Functions -  53 

Section  II.     Differentiation  of  Functions --  5^ 

Sign  of  the  Differential 5^ 

Differential  of  a  Function  Consisting  of  Terms —  5S 

Forms  of  Algebraic  Terms 60 

Differential  of  a  Product  of  Two  Variables 63 

Geometrical  Illustration 65 

Differentials  of  Fractions 68 

Differential  of  the  Power  of  a  Variable 71 

Root        "          "         73 

"                "        Function  of  Another  Function .  74 

Section  III.     Successive  Differentials.. 

vii 


Vm  .     CONTENTS. 

Maclaurin's  Theorem --    85 

Taylor's  Theorem  _ 89 

Identity  of  Principle  in  Both  Theorems  (note) 93 

Section  IV.     Maxima  and  Minima 95 

Method  of  Finding  by  Substitution  — ..  96 

Meaning  of  a  Maximum 97 

Method  of  Finding  by  the  Second  Differential 98 

Use  of  Other  Differential  CoefiBcients gq 

Examples  Illustrating  Different  Cases.. 100 

Analytical  Demonstration  of  the  General   Rule 105 

Section  V.     Application  of  the  Calculus  to  the  Theory  of  Curves.  125 

Definition  of  a  Line 125 

Why  the  Line  Becomes  a  Curve 126 

Direction  of  the  Tangent  Line 127 

Real  Meaning  of  the  Differential  Equation 129 

Sign  of  the  First  Differential  Coefficient _  132 

Section  VI.     Differentials  of  Transcendental  Functions 135 

Differential  of  a'v 135 

"           of  the  Logarithm  of  a  Variable 137 

''        Sine  of  an  Arc   139 

'*        Cosine  of  an  Arc -. —  140 

"               "        Tangent  of  an  Arc... 141 

"               "        Secant  of  an  Arc  -.- 142 

"               "        Versed  Sine  of  an  Arc ■  143 

"               "        Arc 143 

Signification  of  the  Differentials  of  Circular  Functions .  144 

Values  of  Trigonometrical  Lines 14S 

Section  VII.     Tangent  and  Normal  Lines  to  Algebraic  Curves 150 

Length  of  Subtangent  to  any  Curve --  152 

"        "    Tangent             "           " - 153 

"    Subnormal         "            " 153 

"    Normal              "           "     154 

Application  of  the  formulas 1 54- 1 56 

Section  VIII.     Differentials  of  Curves 157 

Differential  Plane  Surfaces  Bounded  by  Curves 15S 

"           Surfaces  of  Revolution _ -.    -    --  162 

"           Solids  of  Revolution 165 

Section  IX.     Polar  Curves 168 

Tangents  to  Polar  Curves -.  168 

Differential  of  the  Arc  of  a  Polar  Curve 171 


CONTENTS.  IX 

Subtangent  to  a  Polar  Curve _ 171 

Tangent  "         "         "       _..    171 

Subnormal      "         "         "       172 

Normal  "         "         " 172 

Surface  bounded  by  a  Polar  Curve 173 

Spirals 173 

Spiral  of  Archimedes _ 1 74 

Hyperbolic  Spiral 176 

Logarithmic    " 179 

Section  X.    Asymptotes _ 182 

How  to  Find  Them. _. 183 

Examples _ 183-186 

Section  XI.     Signification  of  the  Second  Differential 187 

Sign  of  the  Second  Differential  Coefficient 188 

Value    "  "  "  ♦'         190 

Section  Xn.     Curvature  of  Lines 192 

Measure  of  Curvature 192 

Contact  of  Curves 193 

Constants  in  the  Equation  of  a  Curve 195 

Osculatrix  to  a  Curve 199 

Radius  of  Curvature 202 

Section  XIII.     Evolutes 207 

Properties  of  the  E volute 207 

To  Find  the  Equation  of  the  Evolute 210 

Section  XIV.     Envelopes 216 

Definition  of  an  Envelope  . 2x8 

How  to  Obtain  its  Equation 218 

Section  XV.  Application  of  the  Calculus  to  the  Discussion  of  Curves  229 

The  Cycloid 229 

Properties  of  the  Cycloid 230 

Logarithmic  Curve 238 

Section  XVI.     Singular  Points 243 

Maxima  and  Minima 243-248 

Cusj)s 24S 

Conjugate  Points _ 253 

Multiple  Points 254 


CONTENIS. 


PART     II 


INTEGRAL    CALCULUS. 

PAGE 

Section  I.     Principles  of  Integration — _.  261 

Integration  of  Compound  Differential  Functions. .  263 

Monomial           "                     "         __ 263 

Particular  Binomial  Differentials 266 

Rational  Fractions 267 

Between  Limits 277 

by  Series 278 

of  Differentials  of  Circular  Arcs 279 

Section  II.     Integration  of  Binomial  Differentials 282 

Integration  of  Particular  Forms 283-288 

"          by  Parts 291 

Formulas  for  Reducing  Exponents 293-303 

Integration  of  Exponential  Differentials 303 

"                  Logarithmic  Differentials 307 

Section  III.     Application  of  the  Calculus  to  the  Measurement  of 

Geometrical  Magnitudes 311 

Rectification  of  Curves 312 

Quadrature  of  Curves 319 

Surfaces  of  Revolution _ _ 332 

Cubature  of  Solids  _ -  336 


Introduction. 


THE  PHILOSOPHY  OF    THE    CALCULUS. 

In  the  early  history  of  geometry,  among  a  rude  people, 
whose  ideas  were  generally  bounded  by  that,  only,  which 
could  be  seen  and  tested  by  experiment,  the  science  was 
necessarily  confined  to  the  solution  of  the  most  simple  propo- 
sitions. After  the  measurement  of  rectangles,  it  was 
easy  to  proceed  to  that  of  parallelograms,  and  thence  to 
triangles  ;  and  as  all  rectilinear  figures  could  be  divided 
into  triangles,  they  were  able  to  find  the  properties  of  a 
great  variety  of  such  figures  without  difficulty.  Similarly^ 
the  measurement  of  rectangular  parallelopipeds  led  to  that 
of  those  which  were  oblique,  and  thence  to  the  measurement 
of  prisms  with  triangular  and  polygonal  bases.  But  when 
they  came  to  figures  bounded  by  curved  lines,  and 
volumes  bounded  by  curved  surfaces,  the  primitive  meth- 
ods which  they  had  used,  were  found  to  be  powerless; 
and  it  was  not  until  philosophy  began  to  be  cultivated, 
and  men  began  to  look  beyond  the  mere  evidence  of  their 
senses,  and  to  cultivaie  ideas  of  abstract  truth,  that  any 
important  advance  was  made  in  the  science.  At  length, 
some  unknown  philosopher  struck  out  a  new  method  which 
opened  up  a  brilliant  career  to  geometry.  The  exact  period 
of  its  invention  is  not  known,  but  the  first  extensive  use 
of  it  was  made  in  the  works  of  Euclid  and  Archimedes,  who 
rapidly  enlarged  the  boundaries  of  matliematical  knowledge 
and  brought  the  science   of  geometry   to  such   a   degree   of 


INTRODUCTION, 


perfection  that  it  remained  for  two  thousand  years,  without 
further  progress.  The  method  which,  in  their  hands,  worked 
such  wonders  is  called, 

THE  METHOD  OF  EXHAUSTION. 

It  is  also  called  the  "  reductio  ad  absurdum''  because  it 
shows  that  every  supposition  but  the  true  one  leads  to  an 
absurdity. 

*  "As  the  ancients  "  says  Carnot "  admitted  only  demonstrations  which 
are  perfectly  rigorous,  they  believed  they  could  not  permit  themselves 
to  consider  curves  as  polygons  of  a  great  number  of  sides  ;  but  vv^hen 
they  v^ished  to  discover  the  properties  of  any  one  of  them,  they  regarded 
it  as  a  fixed  term,  which  the  inscribed  and  circumscribed  polygons  con- 
tinually approached,  as  nearly  as  they  pleased,  in  proportion  as  they 
augmented  the  number  of  their  sides.  In  this  way,  they  exhausted  in 
some  sort  the  space  between  the  polygons  and  the  curves ;  which, 
without  doubt,  caused  to  be  given  to  this  procedure  the  name  of  the 
method  of  exhaustion^ 

f  "  This  will,  perhaps  be  more  clearly  seen  in  an  example.  Suppose, 
then,  that  regular  polygons  of  the  same  number  of  sides  are  inscribed 
in  two  circles  of  different  sizes.  Having  established  that  the  polygons 
are  to  each  other  as  the  squares  of  their  homologous  lines,  they  con- 
cluded, by  the  method  of  exhaustion,  that  the  circles  are  to  each  other 
as  the  squares  of  their  radii.  That  is,  they  supposed  the  number  of  the 
sides  of  the  inscribed  polygons  to  be  doubled,  and  this  process  to  be 
repeated  until  their  peripherics  approached  ns  near  as  we  please  to  the 
■circumferences  of  the  circles.  Ag  the  cpaces  between  the  polygons  and 
the  circles  were  continually  decreasing,  it  was  seen  to  be  gradually 
■exhausted  ;  and  hence  the  name  of  the  method.  But  although  the  poly- 
gons, by  thus  continuing  to  have  ihe  number  of  their  sides  doubled, 
might  be  made  to  approach  the  circumscribed  circles  more  nearly  than 
the  imagination  can  conceive,  leaving  no  appreciable  difference  between 
them  ;  they  would  always  be  to  each  other  as  the  squares  of  their 
homologous  sides,  or  as  the  squares  of  the  radii  of  the  circumscribed 
circles.  Hence  they  conjectured  that  the  circles  themselves,  so  very 
like  the  polygons  in  the  last  stage   of  their  fullness  or  roundness,  were 

*  Reflexions  sur  la  Metaphysique  du  calcul  infinitesimal. 
+  Bledsoe  —  Philosophy  of  Mathematics. 


METHOD    OF    EXHAUSTION.  5 

to  each  other  in  the  same  ratio,  or  as  '  the  squares  of  their  radii.'  But 
it  was  the  object  of  the  ancient  geometers,  not  merely  to  divine,  but  to 
demonstrate.  A  perfect  logical  rigor  constituted  the  very  essence  of 
their  method.  Nothing  obscure,  nothing  vague,  was  admitted,  either 
into  their  premises,  or  into  the  structure  of  their  reasoning.  Hence  their 
demonstrations  absolutely  excluded  the  possibility  of  doubt  or  contro- 
versy ;  a  character  and  a  charm  which,  it  is  to  be  lamented,  the  mathe- 
matics has  so  often  failed  to  preserve  in  the  spotless  splendor  of  its  primi- 
tive purity. 

"  HaviUj^  divined  that  any  two  circles  (Cand  c)  are  to  each  other  as 
the  squares  of  their  radii  (R  and  r),  the  ancient  geometers  proceeded  to 
demonstrate  the  truth  of  the  proposition.  Tiiey  proved  it  to  be  necessa- 
rily true  by  demonstrating  every  other  possible  hypothesis  to  be  false." 

Thus,  said  they,  the  inscribed  polygons  P  and  p  are  to 
each  other  as  R  " :  r^,  and  if  P  is  not  to  C  as  p :  c  then  let  us 
suppose  P  :  C  :  :  p  :  c' ;  c'  being  a  circle  smaller  than  c.  In- 
crease the  number  of  sides  of  the  polygons,  until  p  becomes 
greater  than  c' ;  (for  we  can  make  p  approach  c  as  near  as  we 
please)  then  we  shall  have  P :  C  : :  p  :  c'  in  which  the  first 
consequent  is  greater  than  its  antecedent,  while  the  second 
consequent  is  less  than  its  antecedent,  which  is  absurd ;  and 
we  can  not  have  P:  C  :  :  p :  a  circle  less  than  c.  By  a  similar 
course  of  reasoning,  using  the  circumscribed  polygons,  it 
may  be  shown  that  we  can  not  have  P:  C::  p:  a  circle 
greater  than  c.  Hence  the  proportion  P  :  C  : :  p  :  c  is  true, 
and  therefore  P  :  p  : :  C  :  c: :  R":  r'".  This  process,  by  which 
every  posssible  supposition,  except  the  one  to  be  demon- 
strated was  shown  to  lead  to  an  absurdity,  has  also  been 
called  the  "  reductio  ad  absurdum^"'  as  well  as  the  "  method 
of  exhaustion. " 

*  "  By  this  m.ethod  the  ancients  demonstrated  all  the  intricate  problems 
in  elementary  geometry,  and  brought  that  science  to  the  condition  in 
which  it  remained  for  two  thousand  years.  Truths  were  waiting  on  all 
sides  to  be  discovered,  and  continued  to  wait  for  centuries,  until  a  more 
powerful  instrument  of  discovery  could  be  invented." 

*  Bledsoe — Philosophy  of  Mathematics. 


6  INTRODUCTION. 

At  length  the  period  of  stagnation  was  ended,  and  in 
1637  the  brilliant  genius  of  Descartes,  seizing  upon  a  new 
idea,  boldly  followed  its  lead  until  he  developed  a  system 
whose  results  astonished  and  delighted  the  world.  This 
system  is  that  of 

ANALYTICAL    GEOMETRY. 

Breaking  away  from  the  idea  of  determinate  values  and 
absolute  conditions,  he  adopted  that  of  dependent  conditions 
and  relative  values,  which  no  longer  fixed  unchangeably  the 
quantities  sought,  but  gave  them  a  wide  range,  so  that  with- 
in certain  limits  they  could  have  all  possible  values.  Hence 
they  were  called  variables^  while  those  quantities  whose 
values  were  fixed  were  called  constants. 

In  every  equation  containing  a  single  unknown  quantity, 
the  value  of  that  quantity  is  absolutely  fixed  by  the  condi- 
tions expressed  in  the  equation.  If  we  have  two  unknown 
quantities,  and  two  equations,  or  sets  of  conditions,  both 
values  are  still  fixed.  If  the  higher  power  of  the  unknown 
quantity  is  involved,  the  number  of  values  is  greater,  but 
they  are  equally  fixed  and  certain.  This  idea  of  fixedness 
of  value  underlies  all  algebraic  operations  of  an  ordinary 
kind. 

Now  suppose  we  have  two  unknown  quantities  in  one 
equation,  with  no  other  conditions  given  than  those  ex- 
pressed by  the  equation  itself.  In  that  case  the  values  of 
both  quantities  are  absolutely  indeterminate.  But  if  we  know 
or  assume  any  specific  value  for  one,  we  can  at  once  deter- 
mine the  corresponding  value  of  the  other;  so  that  while  the 
equation  will  give  the  independent  value  of  neither  quantity, 
it  will  give  the  simultaneous  values  of  both;  and  these  values, 
will  have  a  certain  range  or  locus,  which  is  in  fact  the  true 
solution  of  the  equation  —  the  path,  so  to  speak,  through 
which  the  simultaneous  values  range. 


ANALYTICAL    GEOMETRY.  7 

In  some  equations  the  range  of  values  is  limited  for  both 
variables,  so  that  if  a  value  be  assigned  to  one  beyond  the 
limit,  that  of  the  other  becomes  imaginary;  in  other  cases 
the  value  of  one  only  is  limited,  while  in  others  again  the 
values  of  both  are  absolutely  unlimited;  any  value  of  one 
giving  a  corresponding  real  value  for  the  other. 

Since  the  values  of  these  variables  are  thus  dependent 
on  each  other,  the  equation  expressing  this  dependence  may 
be  considered  as  containing  the  /a7ao(  ihtiv  mutual  relations, 
and  the  fundamental  idea  of  Descartes  was  to  exhil)it  in  his 
equation  the  conditions  or  law  which  confined  the  two 
variables  to  their  prescribed  range  of  values.  This  idea  was 
something  new,  distinct  and  well  defined,  and  a  clear  ad- 
vance beyond  the  methods  of  tlie  ancients. 

But  the  labors  of  Descartes  would  have  been  of  little 
value,  had  he  proceeded  no  farther  than  we  have  indicated. 
In  fact  this  was  but  a  part  of  his  invention,  of  which  the 
specific  object  was  a  method  of  investigating  questions  of 
Geometry.  To  complete  this  purpose,  he  devised  a  new 
and  beautiful  method  of  representing  magnitudes,  to  which 
his  algebraic  equations  could  be  applied.  In  algebra,  all 
values  are  estimated  by  tlieir  remoteness  from  zero.  In 
order  to  make  a  general  application  of  algebraic  symbols  to 
geometry,  it  was  necessary  that  the  value  of  every  line  rep- 
resented by  his  variables  should  be  estimated  from  a  com- 
mon origin  corresponding  with  zero;  and  as  every  point  in 
a  plane  surface  requires  two  values  to  fix  its  position,  two 
such  origins  became  necessary  to  his  system,  in  order  to 
represent  plain  figures  ;  and  these  were  found  in  two  right 
lines,  lying  in  the  plane  of  the  object  to  be  represented,  and 
intersecting  in  a  known  angle  —  generally  a  right  angle* 
From  these  two  lines  all  values  or  distances  to  points  were 
estimated ;  the  positive  on  one  side  and  the  negative  on  the 
other,  of  each  line;  while  for  points  I'/i  the  lines,  one  of  the 
vnlues  would  of  course  be  zero. 


8  INTRODUCTION. 

Having  then  a  method  of  representing  the  position  of  a 
point  by  algebraic  symbols,  it  was  easy  to  apply  his  analysis 
to  the  representation  of  lines,  by  making  the  locus  or  range 
of  simultaneous  values  of  the  variables  to  correspond  with 
the  locus  of  the  points  in  the  line  —  that  is,  with  the  line 
itself. 

Thus  the  method  of  Descartes  was  two-fold  —  the  alge- 
braic idea  of  two  variables  in  one  equation  with  a  range  of 
simultaneous  values,  and  the  geometrical  idea  of  co()rdinate 
representation,  and  these  two  being  adapted  to  each  other, 
united  to  form  a  method  of  investigating,  in  an  easy  and 
simple  manner,  questions  of  geometry  which  had  taxed  the 
utmost  powers  of  the  ancients. 

The  invention  of  Descartes  did  not,  however,  change  the 
conceptions  that  had  been  formed  of  the  nature  and  compo- 
sition of  quantities.  A  rectangle  was  still  one  entire  sur- 
face, and  a  cone  and  sphere  were  still  "solids." 

The  first  important  departure  from  the  old  ideas  about 
magnitudes,  was  when  Kepler  introduced  the  notion  that  all 
magnitudes  are  made  up  of  an  infinite  number  of  infinitely 
small  parts.  In  1635  Cavalieri  modified  this  conception 
somewhat,  and  formulated  it  into  a  system,  which  he  pub- 
lished under  the  title  of  the 

METHOD  OF    INDIVISIBLES.* 

"  In  this  method,  which  opened  a  new  era  in  Geometry,  lines  are  con- 
ceived to  be  made  up  of  points,  surfaces  composed  of  lines,  and  volumes 
composed  of  surfaces." 

This  idea  was  a  startling  one,  and  was  not  received 
without  very  decided  objections.  "These  hypotheses,"  says 
Carnot,  "are  certainly  absurd,   and  ought   to  be  emplo3'ed 

*  In  discussing  the  method  of  indivisibles  the  writer  has  drawn  largely  on  "The 
Philosophy  of  Mathematics,"  by  Dr.  A.  T.  Bledsoe. 


METHOD    OF     INDIVISIBLES.  9 

with  circumspection."  One  would  naturally  inquire,  if  they 
are  absurd,  why  employ  them  at  all  ?  The  reason,  which 
seems  to  have  reconciled  Carnot  to  their  use,  is,  that  they 
produce  true  results.  Hence,  as  if  by  way  of  apology,  he 
says  : 

"  It  is  necessary  to  regard  them  as  means  of  abbreviation,  by  means 
of  which  we  obtain,  promptly  and  easily,  in  many  cases,  what  could  be 
discovered  only  by  long  and  pamful  processes,  according  to  the  method 
of  exhau-.tion."  "  The  great  geometers,  who  followed  this  method,  soon 
seized  its  spirit  ;  it  was  in  great  vogue  with  them  until  the  discovery  of 
the  new  calculus,  and  they  paid  no  more  attention  to  objections  iliat 
were  then  raised  against  it  than  the  Bernouillis  paid  to  those  that  were 
afterwards  raised  against  the  infinitesimal  analysis.  It  was  to  this  method 
of  indivisibles  that  Pascal  and  Roberval  owed  their  profound  re^earches 
concerning  the  cycloid." 

"Thus,"  says  Dr.  Bledsoe, "  while  appealing  to  the  prac- 
tical judgment  of  mankind,  they  treated  the  demands  of  our 
rational  nature  with  disdain,  and  the  more  so,  perhaps, 
because  these  dCiUands  were  not  altogether  silent  in  their 
own  breasts." 

To  assume  the  truth  of  a  theory  because  it  accounts  for 
all  the  facts,  can  not  be  admitted  in  mathematics,  because 
mathematical  truth  must  admit  of  no  doubt  in  the  mind  of 
him  who  comprehends  it;  and  in  this  kind  of  reasoning 
there  is  always,  at  least,  one  source  of  doubt,  namely,  that 
there  may  be  other  theories  that  will,  equally  well,  account 
for  the  same  facts ;  which  is  indeed  eminently  the  case  in 
this  very  subject.  Hence  our  first  enquiry  should  be,  "■  is 
the  theory  based  on  principles,  the  truth  of  which  com- 
mends itself  to  every  man's  consciousness,  and  by  which 
the  theory  can  be  proven  true?  "  "Cavalieri  acknowledged," 
says  Carnot,  "  that  he  could  not  give  a  rigorous  demon- 
stration of  his  method."  He  asserted  however  that  it  was 
simply  a  corollary  from  the  method  of  exhaustion  —  an 
assertion  that  is  not  very  clear  to  an  ordinary  mind. 


lO  INTRODUCTION. 

*"  But  Pascal  himself,  though  universally  recognized  as  one  of  the 
greatest  geniuses  that  ever  lived,  could  not  comprehend  the  hypotheses 
or  postulates  of  the  method  of  indivisibles  as  laid  down  by  Cavalieri. 
Hence,  while  he  continued  to  use  the  language  of  Cavalieri,  he  attached 
a  different  meaning  to  il  —  a  change  which  is  supposed  by  writers 
on  the  history  of  mathematics  to  have  improved  the  rational  basis  of 
the  method.  '  By  an  indefinite  number  of  lines'  said  he  '  he  always 
iTfeant  an  indefinite  number  of  small  rectangles,'  '  of  which  the  sum  is 
certainly  a  plane. '  In  like  manner,  by  the  term  '  surfaces '  he  meant 
'indefinitely  small  solids,'  the  sum  of  which  would  surely  make  a 
solid.  Thus  he  concludes,  '  if  we  understand,  in  this  sense,  the  ex- 
pressions the  sum  of  the  lines ^  the  stem  of  the  planes^  etc.,  they  have 
notliing  in  them  but  what  is  perfectly  conformed  to  pure  geometry." 
This  is  true.  The  sum  of  the  planes  is  certainly  a  plane,  and  the 
sum  of  little'  solids  is  certainly  a  solid.  But  from  this  point  of  view 
it  is  surely  not  proper  to  call  it  '  the  method  of  indivisibles,'  since  every 
plane  as  well  as  every  solid  may  easily  be  conceived  to  be  divided.  The 
improved  postulates  of  Pascal  deliver  us,  indeed,  from  the  chief  difficulty 
of  the  method  of  indivisibles,  properly  so  called,  only  to  plunge  us  into 
another  —  into  the  very  one,  in  fact,  from  which  Cavalieri  sought  to 
effect  an  escape  by  the  invention  of  his  method." 

"  Thus,  if  we  divide  any  curvilinear  figure  into  rectangles,  no  matter 
how  small,  the  sum  of  these  rei:tangles  will  not  be 
exactly  equal  to  the  area  of  the  figure.  On  the  con- 
trary, this  sum  will  differ  from  that  area  by  a  surface  /  p. 
€qual  to  the  sum  of  all  the  little  mixtilinear  figures  at 
the  ends  of  the  rectangles  (Fig,  i).  It  is  evident, 
however,  that  the  smaller  the  rectangles  are  made,  or  the  greater  the 
number  becomes,  the  less  will  be  the  difference  in  question.  But  how 
could  Cavalieri  imagine  that  this  difference  would  ever  become  abso- 
lutely nothing,  so  long  as  the  inscribed  rectangles  continue  to  be  surfaces? 
Hence,  in  order  to  get  rid  of  this  difference  altogether,  and  to  arrive  at 
the  exact  area  of  the  proposed  figure,  he  conceived  the  small  rectangles  to 
increase  in  number  until  they  dwindled  into  veritable  lines.  The  sum  of 
these  lines  he  supposed  would  be  equal  to  the  area  of  the  figure  in  ques- 
tion, and  he  was  confirmed  in  this  hypothesis,  because  it  was  found  to  lend 
to  perfectly  exact  results.  Thus,  his  hypothesis  was  adopted  by  him,  not 
because  it  had  appeared  at    first,  or  in  itself  considered,   as  intuitively 

*Bledsoe — Philosophy  of  Mathematics. 


METHOD    OF    IN  DI  VISIBI.ES.  II 

CL'itain,  but  because  it  appeared  to  be  the  only  means  of  escape  from  a 
false  hypothesis,  and  because  it  led  to  so  many  exactly  true  results.  But 
when  this  hypothesis,  abstractly  considered,  was  found  to  shock  the 
reason  of  mankind,  who,  in  the  words  of  Carnot,  pronounced  it  *  cer- 
tainly absurd,'  the  advocates  of  the  method  of  indivisibles  were  obliged 
to  assume  new  ground.  Accordingly,  they  discovered  that  indivisibles 
might  be  divided,  and  that  by  '  the  sum  of  right  lines  '  was  only  meant 
*  tlij  sum  of  the  indefinitely  small  rectangles.'  Pascal  seems  to  believe, 
in  fact,  that  such  was  the  meaning  of  Cavalieri  himself. 

"  Now  it  is  just  as  evident  that  a  curvilinear  figure  is  not  composed 
of  rectangles,  as  that  it  is  not  composed  of  right  lines.  Yet  Pascal,  the 
great  disciple,  adopted  the  supposition  as  the  only  apparent  means  of 
escape  from  the  absurdity  imputed  to  thnt  of  the  master ;  and  he 
pointed  to  the  perfect  accuracy  of  his  conclusions  as  a  proof  of  the  truth 
of  his  hypothesis.  For,  strange  to  say,  the  sum  of  the  rectangles,  as 
well  as  the  sum  of  the  lines,  was  found  to  be  exactly  equal  to  the  cur- 
vilinear figure.  Whit,  then,  became  of  the  little  mixtilinear  figures  at 
the  extremities  of  the  rectangles  ?  How,  since  they  were  omitted  or 
thrown  out,  could  the  remaining  portion  of  the  surface,  or  the  sum  of 
the  rectangles  alone  be  equal  to  the  whole  ?  Pascal  just  cut  the  Gordian 
knot  of  this  difficulty  by  declaring  that  if  two  finite  quantities  '  differ 
from  each  other  by  an  indefinitely  small  quantity'  then  'one  may  be 
taken  for  the  other  without  making  the  slightest  d'ffcrrence  in  the  result.' 
Or,  in  other  v/ords,  that  an  infinitely  small  quantity  may  be  added  to,  or 
subtracted  from  a  finite  quantity  without  making  the  least  change  in  its 
magnitude.  It  was  on  this  principle  '  that  he  neglected  without  scruple,' 
as  Carnot  says,  '  these  little  quantities  as  compared  with  finite  quantities; 
for  we  see  that  Pascal  regarded  as  smiple  rectangles,  the  trapeziums  or 
little  portions  of  the  area  of  the  curve  comprised  between  the  two  con- 
secutive ordinates,  neglecting  consequently  the  little  mixtilinear  triangles 
which  have  for  their  bases  the  differences  of  those  ordinates.'* 

"  Carnot  adds,  as  if  he  intended  to  justify  this  procedure,  that  'no 
person,  however,  has  been  tempted  to  reproacii  Pascal  with  want  of 
severity.'  This  seems  the  more  unaccountable,  because  Carnot  himself 
has  repeatedly  said  that  it  is  an  error  to  throw  out  such  quantities  as 
nothing.  Nor  is  this  all.  No  one  can  look  the  principle  fairly  and  fully 
in  the  face,  that  an  infinitely  small  quantity  may  be  subtracted  from  a 
finite  quantity  without  making  even  an   infinitely  small  difference  in  its 

*Carnot,  Chap.  III.,  p.  146. 


12 


INTRODUCTION, 


value,  and  yet  regard  it  as  otherwise  than  absurd.  It  is  only  when  such 
a  principle  is  recommended  to  the  mathematician  by  the  desperate  exi- 
gencies of  a  system  which  strains  his  reason,  warps  his  judgment,  ard 
clouds  his  imagination,  that  it  is  admitted  to  a  resting  place  in  his 
mind.  It  was  thus,  as  we  have  seen,  that  Pascal  was  led  to  adopt  the 
principle  in  question  ;  and  it  was  thus,  as  we  shall  see,  that  Leibnitz  was 
induced  to  assume  the  same  absurd  principle  as  an  unquestionable 
axiom  in  geometry." 

"  Now  if,  with  Cavalieri,  we  suppose  a  surface  to  be  composed  of 
lines;  or  a  line  ofpoints,  then  we  shall  have  to  add  points  or  no-magni- 
tudes together  until  we  make  magnitudes.  -Nay,  if  lines  are  composed 
ofpoints,  surfaces  of  lines,  and  solids  of  surfaces,  then  it  is  perfectly  evi- 
dent that  solids  are  made  up  of  points,  and  the  very  largest  magnitude 
is  composed  of  that  which  has  no  magnitude  !  or,  in  other  words,  every 
magnitude  is  only  the  sum  of  nothing  !  On  the  other  hand,  if  we  agree 
with  Pascal  that  a  curvilinear  space  is,  strictly  speaking,  composed  of 
rectangles  alone,  then  we  shall  have  to  conclude  that  one  quantity  may 
be  taken  from  another  without  diminishing  its  value  !  Which  term  of 
the  alternative  shall  we  adopt?  On  which  horn  of  the  dilemma  shall  we 
choose  to  be  impaled?  But  is  it,  indeed,  absolutely  necessary  to  be 
swamped  amid  the  zeros  of  Cavalieri,  or  else  to  wear  the  yoke  of  Pascal's 
axiom?  May  we  not  by  a  recurrence  to  the  true  p:Mnciples  of  mathe- 
matical philosophy  find  a  safe  passage  between  this  Scylla  and  Charybdis 
of  the  infinitesimal  method?" 

We  shall  see  further  on. 

The  following  beautiful  example  will  illustrate  most 
clearly  and  favorably  the  method  of  indivisibles.  It  is  given 
by  Carnot  from  Cavalieri  ^ 
and  Pascal  to  recommend 
that  method  : 


"  Let  A  B  (Fig  2),"  says  he,  ^^ 
"be  the  diameter  of  a  semi-circle,/)/ 
A  G  B ;  let  A  B  F  D  be  the  cir- 
cumscribed rectangle  ;,  C  G  the 
radius  perpendicular  to  D  F  ;  let 
the  two  diagonals  C  D,  C  F  also  be  drawn  ;  and  finally  through  any 
point  ni  of  the  line  A  D,  let  the  right  line  in  up  g  be  drawn  perpen- 
dicular to  C  G,  cutting  the  circumference  of  the  circle  at  the  point  n^ 
and  the  diagonal  C  D  at  the  point  /." 


METHOD    OF     INDIVISIBLES.  1 3 

"Conceive  the  whole  figure  to  turn  around  C  G  as  an  axis ;  the  quad- 
rant of  the  circle  A  C  G  will  generate  the  volume  of  a  semi-sphere  whose 
diameter  is  A  B  ;  the  rectangle  A  D  G  C  will  generate  the  circumscribed 
right  cylinder;  the  isosceles  right-angled  triangle  C  G  D  will  generate  a 
right  cone,  having  the  equal  lines  C  G,  D  G  for  its  height  and  for  the 
radius  of  its  base  ;  and  finally  the  three  right  lines  or  segments  of  a  right 
Vine  m  ^,  ng,pg,  will  each  generate  a  circle,  of  which  the  point  g  will 
be  the  centre. 

"  But  the  first  of  these  circles  is  an  element  of  the  cylinder  ;  the  second 
is  an  element  of  the  semi-sphere,  and  the  third,  of  the  cone. 

"  Moreover  the  areas  of  these  circles  are  as  the  squares  of  their  radii, 
and  as  these  three  radii  can  evidently  form  the  hypothenuse  and  the 
two  sides  of  a  right  angle  triangle,  it  is  clear  that  the  first  of  these  circles 
i^  equal  to  the  sum  of  the  other  two  ;  that  is  to  say,  that  the  element  of 
the  cylinder  is  equal  to  the  sum  of  the  corresponding  elements  of  the 
cone  and  semi-sphere ;  and  as  it  is  the  same  with  all  the  other  elements 
cut  out  by  one  horizontal  plane,  it  follows  that  the  total  volume  of  the 
cylinder  is  equal  to  the  sum  of  the  total  volumes  of  the  semi-sphere  and 
cone.  But  we  know  that  the  volume  of  the  cone  is  equal  to  one-third 
that  of  the  cylinder ;  then  that  of  the  semi-sphere  is  two-thirds  of  the 
volume  of  the  circumscribed  cylinder,  as  Archimedes  discovered." 

We  see  here  that  we  may,  with  equal  propriety,  with 
Cavalieri,  consider  the  elements  as  mere  surfaces,  with  no 
thickness,  in  which  case  we  have  something  made  up  of 
nothing;  or,  with  Pascal,  as  infinitely  thin  cylinders,  in  which 
case  we  must  ignore  the  slender  rings,  surrounding  these 
cylinders,  and  whose  interior  sides,  in  the  one  case,  form  the 
actual  surface  of  the  cone,  and  whose  exterior  sides,  in  the 
other  case,  form  the  surface  of  the  semi-sphere. 

The  following  is  the  true  solution  of  this  proposition : 
Suppose  the  cylinder,  semi-sphere  and  cone  to  be  generated, 
each  by  its  element,  commencing  together  at  the  line  D  F 
and  flowing  upwards  at  equal  uniform  rates,  expanding, 
contracting,  or  remaining  constant,  so  that  each  element 
shall  at  all  times  have  its  circumference  in  the  surface  of 
the  body  it  is  generating.     The  rate  at  which  each  volume 


14  INTRODUCTION. 

is  generated  at  any  moment  will  be  in  proportion  to  the  area 
of  the  generatrix  at  that  moment;  and  since  the  sum  of 
the  generatrices  of  the  semi-sphere  and  cone  is,  at  all  times, 
equal  to  the  generatrix  of  the  cylinder,  the  sum  of  the  mag- 
nitudes generated  by  them  will  be  equal  to  the  magnitude 
of  the  cylinder. 

Behold,  then,  the  solution  which  avoids  both  horns  of 
the  dilemma. 

"  It  is  true  that  this  axiom  of  Pascal  has  high  authority  in  its  favor. 
Roberval,  Pascal,  Leibnitz,  the  Marquis  de  L'Hopital,  and  others,  have 
all  lent  the  sanction  of  their  great  names  to  support  it,  and  give  it  cur- 
rency in  the  mathematical  world.  But  does  a  real  axiom  ever  need  the 
support  of  authority  ? 

On  the  other  hand,  there  is  against  this  pretended  axiom,  as  intrinsic- 
ally and  evidently  false,  the  authority  of  men  equally,  or  more,  cele- 
brated in  the  mathematical  world,  such  as  Berkeley,  Maclaiirin,  Carnot, 
Euler,  D'Alembert,  Lagrange  and  Newton,  whose  names  preclude  the 
mention  of  any  others.  The  very  fact  that  mathematicians  disagree 
proves  that  it  is  not  about  an  axiom,  but  only  about  something  else  which 
has  been  set  up  as  an  axiom.  It  is  indeed  the  very  essence  of  geomet- 
rical axioms  that  they  are  necessary  and  universal  truths,  absolutely  com- 
manding the  assent  of  all,  and  shining  like  stars  above  the  dust  and 
darkness  of  human  controversy." 

The  next  important  movement  in  the  mathematical 
world  was  when  Newton  and  Leibnitz  brought  out  their 
systems,  in  which  the  results  are  essentially  the  same, 
while  the  fundamental  idea  or  philosophy  of  each  is  entirely 
different  from  the  other. 

METHOD    OF    LEIBNITZ. 

In  1684  Leibnitz  first  published  his  Differential  Calculus, 
in  which  he  gave  the  first  system  of  rules  for  the  required 
operations.  His  method  was  formed  on  a  conception  similar 
to  that  of  Pascal ;  viz.  that  all  quantities  were  composed  of 


METHOD    OF    LEIBNITZ.  I5 

infinitesimals.  He  considered  surfaces  l)Ounded  by  curved 
lines  as  only  polygons  whose  sides  were  infinitely  small 
straight  lines,  and  whose  areas  were  composed  of  an  infinite 
number  of  infinitesimal  rectangles,  while  the  tangent  lines 
to  curves  were  the  prolongations  of  the  infinitely  small  po- 
lygonal sides  —  that  solids  were  composed  of  an  infinite  num- 
ber of  infinitely  small  cylinders  or  prisms,  and  that  a  double 
curved  surface  was  in  fact  that  of  a  polyedron  with  infinitely 
small  faces,  which,  being  extended  would  form  tangent 
planes  to  the  surface. 

The  whole  superstructure  of  the  method  of  Leibnitz 
rests  upon  two  assumptions  which  are  thus  described  by 
the  Marquis  de  L'Hopital. 

First.  "  We  demand  that  we  can  take  indifferently  the  one  for  the 
other,  two  quantities  which  differ  from  each  other  by  an  infinitely  small 
quantity  ;  or  (what  is  the  same  thing),  that  a  quantity  which  is  increased 
or  diminished  by  another  quantity  infinitely  less  than  itself,  can  be  con- 
sidered as  remaining  the  same. 

Second.  "  We  demand  that  a  curved  line  can  be  considered  as  the 
assemblage  of  an  infinity  of  right  lines,  each  infinitely  small  ;  or  (what  is 
the  same  thing),  as  a  polygon  with  an  infinite  number  of  sides,  each  infi- 
nitely small,  which  determine  by  the  angles  which  they  make  with  each 
other  the  curvature  of  the  lines." 

But  Leibnitz  carried  his  idea  much  further  than  Pascal, 
for  he  considered  each  of  the  infinitesimals  of  his  system  as 
also  composed  of  an  infinite  number  of  parts,  infinitely 
smaller  than  itself ;  and  these,  again,  composed  of  an 
infinite  number  of  parts,  infinitely  smaller  yet ;  and  so  on 
indefinitely.  He  called  them  infinitesimals  of  the  first> 
second,  third,  and  so  on,  orders,  in  which  each  particle  or 
portion  of  any  order  is  infinitely  less  than  one  of  the  pre- 
ceding order,  and  bore  the  same  relation  to  it  that  the 
infinitesimal  of  the  first  order  bore  to  the  oridnal  function. 


X< 


l6  INTRODUCTION. 

He  indicated  these  infinitesimal  increments,  parts,  or 
differences,  by  the  term  "  differentials,"  and  thus  we  find  in 
his  system,  differentials  of  the  first,  second,  third,  and  so  on, 
orders  ;  which  he  called,  "  first  differential,  second  differen- 
tial," and  so  on. 

In  order  to  illustrate  the  peculiarity  of  the  idea  of 
Leibnitz,  let  us  suppose  a  curve  to  be  composed  of  an 
infinite  number  of  right  lines,  infinitely  short,  which,  for  the 
sake  of  perspicuity  we  will  represent  by  the  polygon  ABC 
DE. 

Let  C  F  and  D  G  (Fig.  3)  be  consec- 
utive ordinates  of  the  curve  and  C  D 
an  infinitesimal  side  lying  between  them. 
The  infinitesimal  difference  D  h  between 
these  ordinates  is  called  the  differential 
of  the  ordinate  at  the  point  C,  and  this, 
divided  by  the  corresponding  differential  / 
C  h  (^or  F  G)  of  the  abscissa,  will  give  "^ 
the  tangent  of  the  angle  C  T  F  which  the 
side  CD,  or  the  tangent  line  C  T,  which  is  a  prolongation 
of  the  side  C  D,  makes  with  the  axis  of  abscissas.  But 
the  position  of  this  tangent  line  does  not  decide  which  way 
the  curve  is  bending,  that  is,  whether  it  is  concave  or  con- 
vex toward  the  axis  of  abscissas.  To  ascertain  this  last 
point;  the  next  infinitesimal  side  D  E  of  the  curve  is  taken, 
and  the  infinitesimal  difference  E  k  between  its  two  ordin- 
ates is  compared  with  the  infinitesimal  difference  D  h 
between  the  ordinates  of  the  preceding  side  C  D,  and  the 
difference  between  these  two  infinitesimal  differences  will 
show  which  way  the  curve  is  bending.  If  this  difference  is 
negative,  the  second  infinitesimal  E  k  is  less  than  the  first, 
and  the  curve  is  concave  toward  the  axis  of  abscissas  ;  if  it 
is  positive,  the  last  infinitesimal  difference  is  greater,  and  the 
curve  is  convex  toward  that  axis. 


METHOD    OF    LEIBNITZ. 


17 


Fig.  4. 


The  difference  between  these  two  infinitesimal  differences 
is  called  the  second  differential  of  the  ordinate  ;  and  as  the 
first  differential  is  infinitely  less  than  the  ordinate  C  F,  so 
the  second  differential  is  infinitely  less  than  the  first. 

Now  this  method  of  differences  when  the  figure  is  actu- 
ally a  polygon  is  reasonable  enough,  but  when  applied  to  a 
real  curve,  represented  by  an  equation,  we  find  that  the 
theory  of  the  many  sided  polygon  will  not  account  for  the 
results. 

Let  us  take  for  example  the  parabola  A  P  P',  (Fig.  4), 
of  which  the  equation  is 
V  =  a:^,  and  suppose  it  to  be 
•composed  of  an  infinite  num-  V 
ber  of  infinitely  small  straight 
lines.  For  the  sake  of  i)er- 
spicuity  let  us  consider  P  P' 
<as  one  ot  these  sides  ;  then 
representing  A  B  by  x,  B  B' 
by  h,  P  B  by  J',  and  P'B'  by/, 
we  shall  have 

y  ■=■  x^  and  y   =  {x  -\-  /^) 
whence  we  derive 


A 


=  2.V  +    //  -^' 


which  represents  the  tangent  of  the  angle  P  F  B.  But  the 
true  tangent  of  the  angle  made  by  the  tangent  line  to  a  para- 
bola like  this  is  equal  to  2x;  hence  the  prolongation  of 
P  P'  is  not  tangent  to  the  curve,  and  the  P  P'  is  not  there- 
fore a  part  of  it.  In  considering  the  curve  as  made  up  of 
straight  lines  then,  we  commit  an  error,  and  this  error  we 
can  correct  only    by    reducing   /i    to    zero   in   tlie  equation 


y 


■y  _ 


/i 


=  2X  -\-  /i ,  and  thus,  bringing  the  two  points    P  and 


P'  together,  we  reduce  the  chord  (or  side  of  the  curve)  PP'  to 


l8  INTRODUCTION. 

zero  also,  at  the  same  time;  so  that  while  one  of  the  postu- 
lates ofde  L'Hopital  leads  him  into  error,  the  other  delivers 
him  from  it,  and  thus  he  is  confirmed  in  both  errors  by  the 
correctness  of  his  results.  "  Two  rays  of  light  sometimes 
make  darkness,  but  it  is  reserved  for  mathematics  to  exhibit 
the  phenomenon  of  two  rays  of  darkness  producing  light." 

It  is  easy  to  show  that  the  error  arising  from  taking  P'  P 
as  a  parr  of  the  curve  is  exactly  compensated  by  throwing 
out  h.     For  if  S  D  is  the  true  tangent   line,  then  instead  of 

P'  E  ,      ,  ,  ,  D  E      P  B  ^       , 

— ^  we  should  have  -:— -7  ^^r-^-  for  the  tangent  of  the  angle 

made  by  the  tangent  line  with  the   axis  of  abscissas.     And 

^^     AB        X         .  PB        y  x'        DE 

smce  S  B= =  -  we  have  ^nr^TT —  —  T7~  ~~BnF  — 

2  2  S  B       ^  jc       YzX       P  E 

P'  E 

2X ;  but  we  found  ^tt  —  2x  -\-  //,  so  that  by   throwing  out 

P'  E       DE       ^     , 
h  we  have  reduced  -zr^pr  to  -— -  and  thus   exactly   compen- 
P  E        P  E 

sated  and  canceled  the  error  arising  from  assuming  that  the 

curve    is    composed   of   straight   lines.     We  may  therefore 

freely  adopt  the  language  of  the  celebrated   Lagrange.     He 

thus  expresses  himself  on  the  subject  in  the  last  edition   of 

his  "  Theorie  des  Fonctioiis  Analytiques.  " 

*'  In  regarding  a  curve,"  says  he,  "  as  a  polygon  of  an  infinite  num- 
ber of  sides,  each  infinitely  small  and  of  which  the  prolongation  is  the 
tangent  to  the  curve,  it  is  clear  that  we  make  an  erroneous  supposition  ; 
but  this  error  finds  itself  corrected  in  the  calculus  by  the  omission  which 
is  made  of  infinitely  small  quantities." 

Now  this  explanation  of  the  reason  for  throwing  out  h  was  evidently 
unknown  to  Leibnitz  himself ;  for  when  questioned  on  the  subject  he 
"presented  "  says  Comte  "  an  explanation  entirely  erroneous,  saying  that 
he  treated  infinitely  small  quantities  as  incomparables  and  that  he 
neglected  them  in  comparison  with  finite  quantities  '  like  grains  of  sand 
in  comparison  with  the  sea' — a  view  which  would  have  completely 
changed  the  nature  of  his  analysis,  by  reducing   it    to   a   mere    approxi- 


iMETHOD    OF    LEIBNITZ,  1 9- 

inalive  calculus,  which,  under  this  point  of  view,  would  be  radically 
vicious.  "  Long  before  M.  Comte,  M.  D'Alembert  had  said  substantially 
the  same  thing,  "  M.  Leibnitz "  says  he,  "embarrassed  by  the  objec- 
tions which  he  felt  would  be  made  to  infinitely  small  quantities,  such 
as  the  Differential  Calculus  considered  them,  has  preferred  10  reduce  his 
infinitely  small  quantities  to  be  only  incotiiparables,  which  ruined  the 
geometrical  exactness  of  the  calculus." 

The  first  to  discover  the  true  reason  for  throwing  out  h^ 
was  Berkeley,  Bishop  of  Cloyne. 

"  For  as  much  "  says  he,  "  as  it  may  perhaps  seem  an  unaccountable 
paradox  that  mathematicians  should  deduce  true  propositions  from  false 
principles,  be  right  in  the  conclusion,  and  yet  wrong  in  the  premises,  I 
shall  endeavor  particularly  to  explain  why  this  may  come  to  pass,  and 
show  how  error  may  bring  forth  truth  though  it  can  not  bring  forth  science.  *' 

He  then  gives  a  demonstration  substantially  similar  to 
the  one  already  given,  and  proving  the  same  thing.  But 
notwithstanding  all  this,  together  with  the  testimony  of  such 
men  as  have  been  named  and  many  others,  a  large  part  of 
the  teachers  of  tlie  calculus  practically  adopt  the  method  of 
Leibnitz  ;  and  when  the  puzzled  pupil  demands  an  explana- 
tion, he  is  told  that  the  principle  is  true  because  the  results 
are  true.  Not  being  able  to  deny  this,  the  pupil  contents 
himself  with  learning  the  use  of  the  analytical  machinery  of 
the  calculus,  until,  after  a  while,  he  forgets  his  doubts  and 
acquiesces  in  the  absurdities  of  the  system. 

A  strong  incentive  to  the  continued  use  of  the  infinites- 
imal method  is,  that  it  is  applicable  to  a  great  variety  of 
cases,  which  other  systems,  heretofore  developed,  are  unable 
to  reach.  The  author  of  this  work  ventures  to  believe  that 
he  has  presented  a  system  of  equal  fecundity,  based  on 
sound  principles,  lying  in  the  beaten  track  of  mathematical 
thought 


20  INTRODUCTION. 

METHOD  OF  NEWTON. 

A  few  years  before  the  publication  of  Leibnitz'  Differ- 
ential Calculus,  Sir  Isaac  Newton  invented  a  process  by 
which  he  obtained  the  same  results,  but  the  philosophy  or 
fundamental  idea  of  his  system  was  wholly  different. 

In  fact  there  are  two  systems,  each  of  which  goes  by  the 
name  of  Newton  s  method.  The  one  generally  in  use  is 
known  as  the  ''method  of  limits."     By  this  method,  in   the 

equation  ■^-—-;^ —  =  2jc  +  ^^5  the  quantity  h  is  supposed   to  be 

reduced  to  absolute  zero;  and  2x  is  said  to  be  the  limit  oi 
the  ratio  of  the  increments  of  the  variables,  but  not  the  ratio 
of  any  part  of  them,  even  infinitesimal.  The  results  thus 
obtained  are  undoubtedly  true,  as  will  be  seen  in  the  case 
already  considered.  As  the  secant  P  F  (Fig.  4)  changes  its 
position  by  the  diminution  of  h  or  B  B,  the  limit  will  be 
reached  when  h  has  become  zero,  P  and  P'  have  come  to- 
gether, P  F  has  become  the  tangent  line  D  S,  and  the  tan- 
gent of  the  angle  made  by  it  with  the  axis  of  abscissas  has 
become  2x  —  the  true  result.  Such  is  in  substance  the  con- 
ception which  Newton  formed  for  the  transcendental  anal- 
ysis:  or,  "more  precisely,  that  which  Maclaurin  and 
D'Alembert  have  presented  as  the  most  rational  basis  of 
that  analysis,  in  seeking  to  fix  and  arrange  the  ideas  of  New- 
ton upon  that  subject."  There  is,  however,  one  difficulty 
about  it  which  has  always  puzzled  the  mathematical  world. 
When  h  becomes  equal   to   zero,   the   first   member  of   the 

y' —  V  o 

equation," — —^  =  2.:r -|- /^,  becomes a   symbol   of  inde- 

h  o 

termination.     The  question  that  naturally  arises  is,  how   is 

the  equation  —  =  2x  to  be  interpreted  .'*     It  is  one  that  has 
o 

not  hitherto  been  answered  with  that  certainty  of  evidence 


METHOD    OF     LIMITS,  2  1 

which  carries  with  it  unhesitating  conviction.  The  most 
plausible  answers  have  been,  in  fact,  only  probable  con- 
jectures.    Some  persons  assume  that   there   is   a   real   ratio 

between   two    zeros ;    others   that    the    expression  —  has  no 

meaning:  others,  that  while  it  has  no  real  value,  it  is  a 
"  symbol  of  a  limit  •" — whatever  that  may  mean  —  others,  and 
these  are  he  most  numerous,  without  attempting  to  answer 
the  question,  or  to  give  a  reason  for  their  proceeding,  boldly 

substitute  -7-  in  the  place  of  —  and  assume  that   2X  is  the 
ax  o 

ratio  between  the  differential  of  the  variable  and  that  of  its 

function.     The   following   is    an   example  of  this  kind,  in  a 

well  known  and  poi)uhir  work,  recently  revised  and  cared  of 

its  former  inconsistencies  : 

*  "  It  must  not  be  supposed  that  the  limiting  value  of  a  ratio  is  a  mere 
approximation;  for,  while  we  speak  of  /z  approaching  ze:o,  we  do  this 
merely  to  aid  our  conception  of  the  terms  of  the  ratio  ;  but  in  finding 
the  value  of  the  limit  of  the  ratio,  we  actually  take  /i  equal  to  zero.  The 
limiting  value  of  the  ratio  thus  found  is,  therefore,  not  approximately  but 
absolutely  exact." 

'Tt  is  important,"  he  says,  "to  have  some  convenient  notation  for 
expressing  the  limiting  ratio  of  the  two  simultaneous  increments.  Now 
if  dx  be  put  for  incr.  x  in  the  limiting  value,  tlien,  in  order  to  maintain  a 
uniform  notation,  we  must  put   cij'   for    incr.  y.     We    shall    then   have 

"^^'  -n.     '^y  1        r  J  1    1     T     •.•  1         rincr.  r 

—  =  2x.       By  -^   we  therefore  understand  the  limitmg  value  01-: ; 

(ix  '    ax  mcr.  x 

or  the  limiting  value  of  the  ratio  of  the  increment  of  ^'  to  the   increment 

of  .r  when  the  increment  of  x  approaches  zero." 

But  why  substitute  — -  for  —since  they  are, in  fact,  iden- 
ilx        o 

tical  in  value?     The  reply  is,  that  the  author  has  use  for  dy 

and  dx,  where  zero  would  not  answer   his   purpose   at   all. 

For  he  says  : 

*  Loomis'  Elements  of  the  Calculus,  revised  edition,  pp  24  and  26. 


22  INTRODUCTION. 

dy 
"  Not  only  does  the  symbol  —  have    an    important   meaning,    but    a 

meaning  may  be  attached  to  the  symbols  dy  and  dx  laken  separately. 
These  symbols  may  be  regarded  as  representing  the  simultaneous  rates 
of  increase  of  a  function  and  of  the  variable  upon  which  it  depends." 

N*w  here  is  a  most  wonderful  piece  of  logic!     We   may, 

of  course,  substitute  for  —  any  two   terms  whose  ratio  is  2x, 

whether  we  call  them  dy  and  c/.v,  or  a  and  <^,  or  m  and  n\  but 
by  what  legerdemain  are  cty  and  dx  transformed  into  quanti- 
ties, not  merely  having  the  constant  ratio,  2x^  but  endowed 
with  the  higher  and  more  complex  qualities  by  which  they 
represent  rates^  which  are  relations  between  time  and  change  ; 
and  this  too  by  the  mere  stroke  of  a  pen,  witliout  the  trouble 
of  a  thought  on  the  subject? 

This  assumption  is  characterized   by   Comte   as   an    un- 
warranted  mixture  of  the  system  of  I^eibnitz  with  that  of 

limits  ;  in  reference  to  which  practice,  he  remarksV 

'^     ( 

*  "Several  continental  geometers,  in  adopting  the  method  of  Nevi^ton 
as  the  more  logical  basis  of  the  transcendental  analysis,  have  partially 
disguised  this  inferiority,  by  a  serious  inconsistency,  which  consists  in 
applying  to  this  method  the  notation  invented  by  Leibnitz  for  the  in- 
finitesimal method,  and  which  is  really  appropriate  to  it  alone.     In  des- 

dy 
ignating  by  -^   that  which  logically  ought,  in  the  theory  of  limits,  to  be 

A  V 
denoted  by  L  — —^  and  in  extending  to  all  other  analytical  conceptions 

this  displacement  of  signs,  they  intended,  undoubtedly,  to  combine  the 
special  advantages  of  the  two  methods  ;  but,  in  reality,  they  have  only 
succeeded  in  causing  a  vicious  confusion  between  them,  a  familiarity  with 
which  hinders  the  formation  of  clear  and  exact  ideas  of  either.  It  would 
certainly  be  singular,  considering  the  usage  in  itself,  that  by  the  mere 
means  of  signs,  it  could  be  possible  to  effect  a  veritable  combination  be- 
tween two  theories  so  different  as  those  under  consideration." 

*  Philosophy  of  Mathematics,  p.  238. 


METHOD    OF     LIMITS.  23 

Dr.  Bledsoe  not  only  repudiates  the  infinitesimal  method, 
but  also  the  use  of  the  signs  dx  and   dy  in   the  method   of 

limits.     He  says,  referring  to  the  expression  — : 

"  It  is  said,  if  we  retain  this  symbol,  our  operations  may  be  embarrassed 
or  spoiled  by  the  necessity  of  multiplying,  in  certain  cases,  both  mem- 
bers of  an  equation  by  zero.  But  the  answer  is  easy.  The  first  differ- 
ential coefficient,  if  rendered  accurate,  always  comes  out  in  the  form  of 

-  ;  but  it  need  not  retain  that  form  at  all.     Whether  we  use  -  or -^   in 
o  o       dx' 

writing  the  differential  equation  of  a  tangent  line  to  the  point  x\y\  we 

shall  have  to  eliminate  -  in  the  one  case,  and  -4-r  in  the  other,  in  order 
o  dx' 

to  make  any  practical  application  of  the  formula.     Now  -  isjust  as  easily 

eliminated  by  the  substitution  of  its  value  in  any  particular  case  as  is 
dy' 

-i~r>  ^rid  besides  its  value  may  be  found  and  its  form  eliminated  by  sub- 
stitution without  any  false  reasoning  or  logical  blunder,  which  is  more 

d\' 
than  can  be  said  for  the  form  -r-r-" 

dx 

This  view  would  greatly  restrict  tlie  operations  of  the 
calculus,  confining  them  to  the  solution  of  the  most  simple 
problems  in  it  and  leaving  those  of  a  more  intricate  kind, 
as  well  as  those  of  the  Integral  Calculus  and  most  of  those 
in  mechanics,  without  the  means  of  solution  altogether.  It 
was  this  inherent  weakness  of  the  method  of  limits  that 
induced  the  earlier  mathematicians  to  assume,  wi'.hout 
giving  reasons  or  explanations,  the  symbols  used  in  the 
infinitesimal  metliod,  endeavoring,  at  the  same  time,  to  con- 
ceal their  essential  identity  by  attaching  a  different  meaning 
to  the  term  "  differential."  This,  of  course,  enlarged  the  scope 
of  their  system,  but  still  left  it  unable  to  cope  with  the  more 
intricate  problems  of  mechanics.  It  is  now,  however,  used 
by  very  many  teachers,  but  hitherto  no  logical  demonstra- 
tion has  been  given  to  justify  the  assumptions  which  con- 
tain the  essence  of  the  system.  It  may  here  be  remarked 
th.it  neither  the   system  under   consideration,  nor  the  pure 


24  'INTRODUCTION. 

method  of  limits,  was  ever  adopted  by  Xe.vton  as  his 
method  of  constructing  the  calculus.  His  only  connection 
with  the  method  of  limits  was,  the  demonstration  of  certain 
principles  applicable  to  that  method,  but  which  he  used  for 
a  very  different  purpose.  Nevertheless,  this  compound 
system,  which  has  been  so  unceremoniously  adopted,  pro- 
duces true  results  ;  and,  notwithstanding  the  hitherto  lack 
of  rigid,  mathematical  proof  of  its  soundness,  it  has  at  the 
bottom  a  profound  truth,  which,  being  found,  will  be  a  lamp 
to  our  feet  to  guide  us  out  of  the  region  of  doubt  and  per- 
plexity enveloped  in  the  mists  of  metaphysical  speculation, 
into  the  pure  light  of  reason  and  logical  certaint}'.  This 
principle  is  that  which  lies  at  the  foundation  of  Xewton's 
true  system,  which  he  invented  in  the  year  1666,  and  which 
was  afterwards  published  under  the  title  of  fluxiojis^  This 
method  we  will  examine  in  due  time  ;  but  meanwhile  we  will 
proceed  to  show  what  this  principle  is,  and  how  it  applies 
to  the  elucidation  of  the  method  under, consideration. 

PRIME    AND    ULTIMATE    RATIOS. 

One  of  the  conditions,  under  which  the  variable  quanti- 
ties used  in  the  calculus  are  supposed  to  exist,  is,  that  they 
are  not  only  capable  of  assuming  different  values,  but  are 
actually  in  a  state  of  change.  The  intensity  of  this  state  at 
any  instant  is  measured  by  the  rate  at  which  the  variable 
may  be  changing  at  that  instant ;  and  the  symbol  which 
measures  this  rate,  at  any  moment,  is  the  change  that  would 
take  place  in  a  unit  of  time,  if  the  rate  were  to  continue 
uniform,  from  that  moment,  during  that  unit  ;  and  the  ratio 
of  two  rates  is  the  ratio  of  the  simultaneous  changes  that 
would  take  place,  if  the  rates  were  to  continue  uniform  for 
afiy  length  of  time.  It  is  this  state  or  rate,  as  represented  by 
the  symbol  just  mentioned,  and  not   any  value,  nor   actual 


PRIME    AND    ULTIMATE    RATIOS.  25 

change  of  value,  in  the  quantity,  which  we  represent  by  its 
differential. 

Another  of  the  peculiar  conditions  attached  to  variables 
is,  that  they  are  susceptible  of  two  opposite  relations  to  the 
point  from  which  their  value  is  estimated  —  that  is,  from  the 
zero  point  :  they  may  be  essentially  negative  or  positive, 
independent  of  the  algebraic  sign  which  precedes  them  ;  and 
the  same  quantity  may  pass,  continuously,  from  a  positive 
to  a  negative  state,  or  vice  versa,  through  the  zero  point 
which  divides  them,  without  losing,  at  that  point,  all  its 
characteristics  as  a  quantity.  This  point  of  separation  is 
generally  arbitrary  ;  as  the  zero  point  in  latitude,  the  axes 
of  coordinates  in  geometry,  etc.,  etc.  Now  the  rate,  at 
which  any  such  quantity  may  be  changing,  is  not  at  all  af- 
fected when  its  value  becomes  zero,  as  it  passes  from  an 
essentially  positive  to  an  essentially  negative  state,  or  vice- 
versa  j  but  will  be  governed  by  the  same  law  at  that  point 
as  that  which  controls  it  elsewhere.  Thus  the  cosine  of  an 
angle  of  120°  is  essentially  negative,  independent  of  the  sign 
which  precedes  it ;  and  in  its  change  from  the  point  where 
the  angle  begins,  it  will  diminish  from  positive  radius  to 
zero,  and  then  become  negative,  while  the  rate  of  change, 
if  determined  by  a  uniform  increment  of  the  angle,  will  be 
greatest  at  the  point  where  the  cosine  itself  is  nothing. 

Rate  is  a  complex  idea  of  which  the  elements  are  time 
and  change  ;  but  when  one  uniform  rate  is  compared  with 
another,  without  regard  to  absolute  values,  the  idea  of  time 
may  be  left  out  of  the  question,  and  the  comparison  may  be 
made  by  that  of  their  simultaneous  changes  ;  so  that  the 
ratio  of  the  simultaneous  increments  of  any  two  variables 
which  increase  at  constant  rates,  does  in  fact  perform  a 
double  office  ;  representing  not  only  the  ratio  of  their  changes, 
but  also  that  of  their  rates  of  change,  which  will  of  course  be 
constant. 


26  INTRODUCTION. 

To  illustrate  this  let  us  take  a  very  simple  case  : 
Let  y  =  ax.     Then  by  giving  to  x  an  increment,  //,  and 
denoting  by  j''  the  increased  value  ofy,  we  have 

y'  =  ax  +  ^^i, 

and  subtracting  and  dividing  by  /i,  we  have 

y  —  y 

■^         ^  =1  a. 


h 

If  now  we  suppose  x  to  be  increasing  at  a  uniform  rate, 
then /^  andy  —  y  will  be  the  uniform  simultaneous  incre- 
ments oi y  and  jv,  and  will  express  their  relative  rates  of  in- 
crease ;  and  these  rates  being  constant,  their  ratio  will  be 
constant,  and  equal  to  the  constant  ratio  {a)  of  the  incre- 
ments, irrespective  of  any  particular  value  of  h.  Now,  if 
we  suppose  h  to  decrease,  the  value  of  the  ratio,  expressed 
by  the  second  member  of  the  equation,  will  not  change, 
even  when  h  becomes  zero  and    the   fraction  assumes   the 

form  — .     How  then  are  we  to  interpret  the  expression 
o 

-  =  «.? 
o 

Evidently,  since  a  has  been   the   ratio,   not   only  of  the 

increments,  but  also  of  their  rates  of  increase,  and  since  we 

have  seen  that  the   rates    do    not  necessarily  vanish  with 

the  values  of  the  quantities,  the   constant   ratio,  <2,  must  be 

that  of  the  rates  of  change  in  the   increments,  as  tjiex_are 

passing  through  the  zero  point   of  their  value.     Hence  the 

expression  — ,  which  is  commonly  supposed  to  be  meaning- 
o 

less,  and  which,  when  isolated,  is  so  in  fact,  has,,  when  it 
arises  as  the  result  of  an  analytical  process,  a  very  definite 
and  distinct  meaning,  which  is  to  be  derived  from  the  con- 
ditions which  produce  it.     Hence  we  infer  that  —  derived 

o 


PRIME    AND    ULTIMATE    PATIOS.  27 

from    the    equation  ■^'    ~  ■^'  —  a   means   that  the   quantities 

h 

represented  by  the  terms  of  the  fraction  are  then  passing 
through  the  point  7vJic  re  they  ivould  change  from  positive  to  nega- 
tive, and  that  while  their  values  have  disappeared,  their 
rates  of  change  are  not  at  all  affected  ;  and  since  a  rate  of 
change  implies  a  changing  (piantity,  we  can  not  consider 
these  zeros,  thus  derived,  as  absolutely  nothing  in  all  respects  ; 
but  while  they  have  no  actuatvalue^  as  quantities,  they  have 
a  character,  and  must  be  treated  as  quantities  in  reference 
to  the  characteristics  which  remain  to  them  when  \.\\it\x  value 
has  departed  —  and  these  characteristics  are,  their  state  of 
change^  and  rate  of  change  j  and  a  must  be  considered  as  the 
ratio  of  their  rates  of  change  at  that  point,  as  well  as  when 
it  was  also  the  ratio  of  their  values. 

r 

y  —  v 

Let  us  now  resume  the  equation  — =  2X  +  h.      In 

/i 

this  case  the  terms  of  the  fraction  do  not  change  uniformly, 
and  therefore  do  not  express  their  respective  rates  of  change  ; 
so  that  this  equation  can  not  be  interpreted  as  in  the  last 
case.  The  ratio  2X  +  h  must,  however,  be  a  function  of  the 
rates,  since  it  depends  upon  them  for  its  value;  let  us  ex- 
amine the  conditions  of  that  dependence. 

We  have  seen  that,  when  we  reduce  h  to  zero  and  obtain 

—  =z  2x,  which  is  the  ratio  with  which  the  increments 
o 

vanish,  the  fraction  is  not  necessarily  an  unmeaning  form  of 
expression ;  and  hence  we  have  a  right  to  look  for  some 
definite  and  rational  interpretation  of  it  —  in  other  words  to 
seek,  with  the  hope  of  an  intelligent  answer,  of  what  quan- 
tities is  2X  the  ratio  ? 

In  discussing  this  subject  our  first  proposition  is,  the  ratio 
with  which  the  increments  vanish  is,  also,  that  with  which  they 
begin  to  be. 

This  proposition  is  so  nearly  self-evident  tiiat   Sir   Isaac 


28  INTRODUCTION. 

Newton  did  not  hesitate  to  assume  it  without  a  demonstra- 
tion. It  may,  however,  be  made  more  apparent  by  a  little 
variation  of  the  process  we  have  been  using.  In  the  equa- 
tion j-  =  .T",  let  X  be  dec?'cased  hy  the  quantity  /^  and  then  let 
it  increase  from  x  —  h  to  x  -\-  h  {Ji  changing  its  sign  at  the 
zero  point)  and  we  shall  have  these  three  forms  of  the  equa- 
tion, viz.  y"  ^  {x  —  Jif^y^x'  and  y  =  {x  -[-  Jif  from  which 
we  derive. 

n 
=:2JC  —  //!,  (l.) 


O 

—  =   2X. 
O 


(2.) 


^L_2'  =  2X  +   /l.  (3.) 

As  /i  diminishes  in  equation  (i),  the  ratio  of  the  decrements 
will  increase,  until  /i  vanishes,  andj^"  being  now  equal  to  _v, 
we  have  equation  (2).  Here  /i  changes  its  sign  and  begins 
to  increase,  completing  the  continuous  increment  of  x  from 
X  —  /i  to  X  -\-  /i  and  producing  equation  (3). 

The  equation—  =2j\;is  evidently  derived  from  the  ratio 
o 

of  the  d^'Crements,  just  as  it  was  from  that  of  the  increments,, 
and  is  in  fact  the  point  where  the  former  end  and  the  latter 
begin.  Hence,  in  equation  (2),  we  may  consider  zx  as  the 
ratio  with  which  the  increments  begin. 

Our  second  proposition  is,  that  the  ratio  —  =2.t  is7?o^th& 

ratio  of  any  part  of  the  increments,  /i  and  j'  — y.  It  is  just 
here  that  we  meet  and  deny  the  infinitesimal  theory,  which 
is  that  2x  is  the  ratio  of  the  last  values  of  the  increments  as 
they  vanish — that  is  to  say  it  is  the  ratio  of  the  infinitely 
small  portions  of  the  increments.  If  authority  were  of  any 
avail   in   such  a   case   as  this,   we  have   that   of  Sir  Isaac 


PRIME    AND    ULTIMATE    RATIOS.  29 

Newton  most  directly  and  emphatically  upon  this  very  point.* 
But  no  authority  is  needed  to  establish  the  truth,  that  zx  -\-  h 
is  not  equal  to  2x  as  long  as  h  is  anyihmg — we  ;;wj"/ think  so 
if  we  think  at  all.  In  fact  the  infinitesimal  theory  would 
prol)ably  never  have  been  thought  of,  had  it  not  seemed  to 
be  necessary  to  account  for  the  ratio  2x,  by  supposing  that 
there  nmst  be  some  sort  of  quantities  to  be  compared,  and 
infinitesimals  were  invented  to  supply  the  demand. 

The  theory  can  no  longer  be  admitted  on  tliis  ground, 
for  we  have  seen  that,  in  a  similar  case,  the  zeros  are  not 
absolute  nonentities.  They  have  indeed  lost  one  of  the 
characteristics  of  quantity,  viz.,  that  of  value^  but  have  re- 
tained those  of  change  and  rate  which  will  readily  furnish 
quantities  to  be  compared,  and  thus  leave  the  infinitesimal 
theory,  with  all  its  inconsistencies  and  absurdities,  without 
the  shadow  of  a  reason  for  its  existence. 

We  now  come  to  the  question  —  of  what  is  2x  the  ratio.'* 
The  increments  must,  of  course,  begin  with  the  same  rates 
of  increase  as  those  with  which  the  variables  themselves 
were  increasing  at  that  moment,  and  before  the  increments 
had  acquired  any  magnitude ;  and  these  rates  will  be  repre- 
sented by  the  increments  that  would  take  place  in  any  unit  of 
time  should  the  rates  continue  to  be  uniform.  But  if  the 
rates  were  to  continue  uniform,  their  ratio  as  well  as  that  of 
the  increments  would  be  constant,  and  would  be  that  with 
which  the  increments  began,  and  with  which  the  variables 
were  increasing  at  the  same  moment-.  It  is  true  that  in  the 
case  under  consideration  the  increments  are  not  formed  at 
a  uniform  rate,  and  their  ratio,  therefore,  is  not  constant, 
and  is  not  that  of  their  rates  of  change;  but  it  is  also  true 
that  2X  is  ivhat  would  be  their  constant  ratio  if  they  did  in- 
crease uniformly  from  the  beginning;  and  these  suppositive 
uniform  increments,  as  we  have  seen,  are   the  very  symbols 

*Priiicipia,  Book  I,  Lemma  IX.     Scholium. 


3©  INTRODUCTION. 

tvhich  meastire  the  rates  with  which  the  increments  bezin  • 
or  what  is  the  same  thing,  the  rates  with  which  the  variables 
were  increasing  at  the  same  moment,  whatever  changes  the 
rates  may  undergo  afterwards.  Hence  2x  —  the  ratio  with 
which  the  increments  begin —  is  also  the  ratio  of  the  supposi- 
tive  uniform  increments  or  symbols  which  measure  the  rates 
of  change  in  x  and  y  (or  x'-)  at  the  moment  the  increments 
begin,  and  is  therefore  the  ratio  of  the  rates  them- 
selves. 

The  abstract  proposition  may  be  thus  demonstrated. 

Prime  and  Ultimate  Ratios. 

1.  The  ratio  of  uniform  increments  is  that  with 
which  they  vanish  and,  also,  begin;  and  is  also  the 
ratio  of  their  rates  of  increase. 

2.  The  ratio  with  which  variable  increments 
vanish  is  that  with  which  they  begin  ;  and  is  also 
the  ratio  of  what  would  be  the  increments  if  they 
were  made  at  a  uniform  rate. 

3.  These  suppositive,  simultaneous,  uniform  incre- 
ments ARE  SYMBOLS,  WHICH  represent  the  initial 
rates  of  increase  of  the  real  increments. 

4.  Hence  the    ultimate  ratio  (which  is  also   the 

NASCENT  ratio)  OF  THE  INCREMENTS  IS  THE  RATIO  OF  THE 
SYMBOLS  WHICH  REPRESENT  THE  RATES  WITH  WHICH  THE 
INCREMENTS  BEGIN  I  AND  THESE  ARE  THE  RATES  AT  WHICH 
THE  VARIABLES  THEMSELVES  ARE  CHANGING  A.T  THAT  MO- 
MENT :      THAT  IS     TO     SAY  THEY     ARE    DIFFERENTIALS     OF 

THOSE  VARIABLES;  AND  THE  ULTIMATE  RATIO  OF  THEIR 
VANISHING  INCREMENTS  IS  TJii  IR  TRUE  DIFFERENTIAL  CO- 
EFFICIENT. 

If  now  we  represent  these  indeterminate  symbols  of  the 
rates,  by  the  signs  dx  and  dy  (calling  them  differentials  of  x 


PRIME    AND    ULTIMATE    RATIOS.  ^I 

and  of  y)  we  may  substitute  these  terms  in  the  fraction  and 

write  i-  =  2j£:insteaaoi  —  =  2:c,  an  equation  which  is  always 
ax  o 

true,  expressing,  not  the  actual  values  of  the  terms  of  the 

fraction,  but  the  relation  existing  between  the  differential,  or 

rate  of  change  in  x^  and  the  corresponding   rate  of  change 

in_;'  or  X'. 

We  have  therefore  a  right,  notwithstanding  the  opinions 

of  Comte  and  Dr.  Bledsoe,  to  substitute  -7—  tor  —  in  what  is 

ilx  o" 

called  the  method    of   limits,  provided    we    understand  by 

them,  not  infinitely  small  quantities  nor  zeros,  but  symbols^ 

which  represent  the  simultaneous  rates  of  change  '\\\y  and  x^ 

and  whose  absolute  values,  like  those  of  other   variables,  are 

indeterminate  until  we  introduce  the  conditions  necessary 

to  fix  them. 

We  may  take  another  view  of  the  subject  which  will  lead 
to  the  same  result. 

When  any  variable  is  in  a  state  of  change  at  a  variable 
rate,  we  may  always  consider  the  actual  change  occurring 
during  any  unit  of  time,  as  arising  from  two  distinct  causes  ; 
namely,  partly  from  the  rate  existing  at  the  beginning  of  the 
time,  and  partly  from  the  change  in  the  rate  taking  place 
during  that  time.  So  if  a  body  is  falling  to  the  earth,  the 
space  passed  over  during  one  second,  is  composed  of  two 
distinct  parts;  one  being  that  which  it  would  have  passed 
over  at  a  uniform  rate  by  virtue  of  its  initial  velocity  at  the 
beginning  of  the  time,  and  which  would  be  the  measure  of 
that  velocity  ;  and  the  other,  that  passed  over  by  virtue  of 
the  ificreased  velocity  produced  by  the  action  of  the  force  of 
gravity.  Hence  if  we  take  the  ratio  of  the  simultaneous 
increments  of  two  variables  increasing  at  variable  rates, 
this  ratio  will  be  affected  by  the  same  causes  that  control 
the  values  of  the  increments  themselves;  namely,  partly  by 


32  INTRODUCTION 

the  rates  existing  when  tne  increments  began,  and  partly  by 
the  subsequent  changes  in  those  rates  ;  and  any  change  in 
the  ratio  of  the  increments  will  be  due  wholly  to  the  change 
in  the  rates  which  may  take  place  while  the  increments  are 
forming.  For  if  the  increments  were  the  result  of  only  the 
rates  existing  at  their  beginning,  they  would  be  made  at  a 
uniform  rate,  and  their  ratio  would  be  constant,  and,  of 
course,  would  be  the  ratio  of  the  rates  with  which  they 
began  and  continued  ;  but  if  certain  causes  should  change 
the  rates  of  these  increments,  it  would  ordinarily  be  mani- 
fested by  a  change  in  their  ratio. 

Now,  as  we  diminish  the  increments,  we  diminish  the 
changes  in  their  rates,  and  finally  when  they  disappear,  the 
changes  in  their  rates  also  disappear,  —  and  of  course  the 
changes  in  their  ratio  vanish  at  the  same  time,  leaving  it 
just  as  if  there  had  been  no  changes  in  the  rates  —  that  is,  it 
will  be  the  ratio  of  the  rates  with  which  the  variables  were 
increasing  when  the  increments  began,  and  before  any 
changes  in  the  rates  occurred. 

Thus  in  the  equation-^^^ ^=  2x-\-h,  the  quantity  h  in 

h 

the  second  member  is  the  change  in  the  ratio  produced  by 
change  in  the  rate  of  y ;  and  if  we  reduce  the  incre- 
ments to  zero  the  changes  in  the  ratio  will  disappear  and 
the  ratio  2X  will  be  the  same  as  if  x  and  y  were  increasing 
at  a  uniform  rate  while  the  increments  were  forming,  and  is 
therefore  the  ratio  of  their  rates  of  increase  with  which  they 
begin. 

Thus  we  see,  the  truth  of  the  method  of  ultimate  ratios 
(commonly  called  the  method  of  limits)  is  shown  to  rest 
wholly  on  the  doctrine  of  rates  represented  by  concrete 
symbols,  or  suppositive  uniform  increments ;  and  these  are 
what  are  known  as  differentials.  This  being  so  it  follows 
that  the  differentials  of  the  calculus   are   not  infinitesimals, 


PRIME    AND    ULTIMATE    RATIf>S.  ^^^ 

but  ra/es  oi  change  symbolically  expressed,  and  that  the  cal- 
culus itself  has  nothing  to  do  with  infinities  nor  metaphysics 
of  any  kind,  but  is  simply,  the  science  of  rates. 

This  method  of  final  ratios  although  logically  true  is  yet 
very  inferior  to  the  direct  method  of  rates,  for 

Firsts  It  is  indirect.  That  is  never  so  satisfactory  as  a 
direct  philosophical  method  ;  and  when  such  a  method  can 
be  found  (as  it  has  been  in  this  case;,  if  other  things  are 
equal,  it  is  always  to  be  preferred. 

Second^  But  other  things  are  not  equal,  but  are  strongly 
in  favor  of  the  direct  method.  The  method  of  limits,  re- 
quiring an  addition  and  then  a  diminution  of  increments, 
imposes  on  the  mind  the  conception  of  a  process  utterly 
useless,  besides  producing  a  false  conception  of  the  true 
nature  of  a  variable  quantity  and  of  the  philosophical  relation 
between  it  and  its  differential  ;  for  the  process  completely 
ignores  the  essential  characteristic  of  a  true  variable. 

Third.  This  method  is  powerless  to  solve  many  problems 
which  have  hitherto  been  solved  only  by  the  use  of  infini- 
tesimals, whereas  the  direct  method  is  fully  equal  in  scope 
to  the  infinitesimal  system,  its  analytical  processes  being 
generally  the  same,  while  the  idea  suggested  by  those  pro- 
cesses, instead  of  being  absurd  and  false,  is  true  and  phil- 
osophical. Thus,  instead  of  saying,  the  infinitesimal 
increment  of  a  surface  is  equal  to  the  infinitesimal  incre- 
ment of  X  multiplied  by  the  ordinate  7,  we  may  say  the 
rate  of  change  of  the  surface  is  equal  to  the  rate  of  change 
of  :i:  multiplied  byj-.  In  either  case  we  have  ds  ^=  ydx.  The 
first  idea  is  incomprehensible,  while  the  latter  is  plain  to  the 
dullest  comprehension. 

Fourth.  The  demonstration  required  to  prove  the  truth 
of  the  method  of  final  ratios  is  by  no  means  an  easy  one  to 
comprehend  ;  and  the  method  itself  fails  to  keep  prominent- 
ly before  the  mind  the  idea  of  the  differential  as  represented 


34  INTRODUCTION. 

by  the  concrete  symbol,  and  leaves  the  student  to  feel  that 
he  is  working  in  a  region  where  all  is  shadowy  and  unsub- 
stantial. It  is  like  traveling  over  a  rough  road  in  the  dark, 
guided,  it  is  true,  by  an  unerring  hand,  compared  with 
traveling  over  the  same  road  in  the  broad  light  of  day. 

The  boast  of  the  infinitesimal  system  is,  that  it  "presents 
incontestably,  in  all  its  applications,  a  very  marked  superi- 
ority, by  leading  in  a  much  more  rapid  manner,  and  with 
much  less  mental  effort,  to  the  formation  of  equations  be- 
tween auxiliary  magnitudes."* 

This  arises  from  the  fact  that  the  method  is  direct.  In- 
stead of  seeking,  by  the  method  of  final  ratios,  the  differential 
coefficient,  and  finding  from  that  the  differential  itself,  the 
infinitesimal  method  seizes  the  differential  at  once  as  an  in- 
finitesimal increment ;  making  the  differentials  of  all  curves 
to  be  infinitesimal  straight  lines ;  the  velocity  of  variable 
motion  to  be  uniform  during  an  infinitesimal  portion  of  time, 
while  the  body  moves  over  an  infinitesimal  portion  of  space. 

Hence,  since  in  uniform  motion  we  have  5"  =  7'/,  the 
equation  will  hold  good  for  variable  motion   when   the   time 

ds 
and  space  are  infinitesimal,  and  we  have  ds  =  vdf,  or  v=—7- 

dt 

which  is  thus  obtained  directly  with  perfect  ease. 

Now  these  advantages  of  the  infinitesimal  method  are 
inconteslible ;  and  were  the  principles  of  that  method  true, 
there  would  be  nothing  left  to  be  desired.  But  its  principles 
are  false.  No  curve  is  made  up  of  straight  lines  —  no  ve- 
locity is  uniform  in  a  variable  motion  ;  and  hence  if  a  method 
can  be  found  which  is  equally  applicable  to  every  variety  of 
investigation,  equally  direct,  and  requiring  no  more  mental 
power  for  its  conception,  and  yet  rigidly  sound  in  its  princi- 
ples and  logical  in  its  application,  there  should  be  no  hesi- 
tation in  adopting  it. 

*  Comte  Phil,  of  Math  ,  p.  113. 


THE  TRUE  METHOD  OF  NEWTON.  35 

\ow-  the  method  of  rates,  expressed  by  uniform  symbol- 
ical increments,  is  such  a  method.  The  analytical  machinery 
is  preciselv  the  same  as  that  of  the  infinitesimal  method, 
and  it  differs  from  that  system  only  in  the  conception  formed 
of  the  nature  of  the  differential.  The  latter  method  con- 
ceives it  to  be  an  aciiiaL  infinite  si  tnal,  uniform  increment  of 
the  variable,  while  the  former  conceives  it  to  be  a  suppositive, 
imaginary,  symbolic^  iifiiforfu  increment,  which  performs  ex- 
actly the  same  duty.  Thus  instead  of  velocity  being  repre- 
sented by  an  infinitely  small  space  passed  over  in  an  infinitely 
sniall  portion  of  time,  it  is  represented  by  a  suppositive 
space  that  would  be  passed  in  a  suppositive  unit  of  time,  if 
the  velocity  were  to  continue  uniform  during  that  unit; 
which  is  in  fact  the  method  by  which  we  always  practically 
measure  velocity ;  as  when  we  say  ''  a  body  is  falling  to  the 
earth  with  a  velocity  of  50  feet  in  one  second."     In  this  case 

the  suppositive  space  and  suppositive  time  are  the  differen- 

7, 

tials,  and  we  have,  as  in  the  other,  ds  =  idt  or  v  =  — -. 

at 

The  True  Method  of  Xewton. 

I  have  said  that  the  method  of  arriving  at  the  differential 
coefficient  by  means  of  the  ultimate  ratios  of  the  increments, 
or,  in  other  words,  the  method  of  limits,  has  generally  been 
ascribed  to  Sir  Isaac  Xewton  ;  but  this  is  evidently  an  error. 
The  theory  on  which  that  method  is  founded  is  certainly  his, 
and  it  is  but  just  that  he  should  be  held  responsible  for  the 
results  tltat  legitimately  flow  from  it.  But  it  is  not  the  theory 
on  which  he  formed  his  method  of  fluxions.  That  is  con- 
tained in  the  second  lemma  of  the  second  book  of  his  Prin- 
cipia.  In  a  scholium  to  that  lemma  he  says:  "In  a  letter 
of  mine  to  Mr.  J.  Collins,  dated  Dec.  lo.  1672,  having 
described  a  method  of  tangents — which  at  that  time  was 
made  public,  I  subjoined  these  words.    This  is  on: particuar 


;^6  INTRODUCTION. 

or  7'ather  corollary^  of  a  general  method,  zvhich  extends  itself^ 
witJiotit  any  troublesoine  calculation,  not  only  to  the  drawing  of 
tangents  to  any  curved  lines,  whether  geometrical  or  7necha)iical^ 
or  ajiyhoiu  resolving  other  abstruse  kinds  of  problems  about  the 
crookedness  [curvature^  areas ^  lengths,  centers  of  gravity  of 
curves^  etc. ,  nor  is  it  liinited  to  equations  which  are  free  from 
surd  qua?itities.  This  method  I  have  interwoven  with  that  other 
of  worki?ig  equations,  by  reducing  them  to  infinite  se7-ies.  So 
far  that  letter.  And  these  last  words  relate  to  a  treatise  I 
composed  on  that  subject  in  the  year  167 1.  The  founda- 
tion of  that  general  method  is  contained  in  the  preceding 
lemma.  " 

Here  it  is  distinctly  stated  by  Newton  himself  that  he 
had  invented  a  general  method  which  was  applicable  not  only 
to  the  drawing  of  tangents,  but  to  all  the  higher  and  more 
delicate  problems  which  appear  in  the  Differential  Calculus, 
and  that  this  general  method  has  the  lemnia  in  questiojifor  its 

FOUNDATION. 

We  have  then  but  to  examine  this  lemma  to  ascertain 
the  real  basis  on  which  the  "  method  of  Newton  "  was  con- 
structed. For  this  purpose  we  give  the  lemma  in  the  author's 
own  words. 

LEMxMA  II. 

"  The  moment  of  any  genitum  is  equal  to  the  moments  of  each  of  the 
generating  sides  drawn  into  the  indices  of  the  powers  of  those  sides,  and 
into  their  coeffi-cients  contimially. 

"  I  call  any  quantity  a  genitum  which  is  not  made  by  the  addition  or 
subduction  of  divers  parts,  but  is  generated  or  produced  in  arithmetic 
by  the  multiplication,  division  or  extraction  of  the  root  of  any  terms 
whatsoever  ;  in  geometry  by  the  invention  of  contents  and  sides,  or  the 
extremes  and  means  of  proportionals.  Quantities  of  this  kind  are  pro- 
ducts, quotients,  roots,  rectangles,  squares,  cubes,  square  and  cubic  sides 
and  the  like. 

"  These  quantities  I  here  consider  as  variable  and  indetermined,  and 
increasing  or  decreasing  as  it  were  by  a  perpetual  motion  or  flux  ;  and  I 


THE  TRUE  METHOD  OF  NEWTON.  37 

understand  their  momentaneous  increments  or  decrements  by  the  name 
of  moments  :  so  that  the  increments  may  be  esteemed  as  additive  or 
affirmative  moments,  and  the  decrements  as  subducted  or  negative  ones. 
But  take  care  not  to  look  upon  finite  particles  as  such.  Finite  particles 
are  not  moments,  but  the  very  quantities  generated  by  the  moments.  We 
are  to  conceive  them  as  the  just  nascent  principles  of  finite  magnitudes. 
Nor  do  we  in  this  lemma  regard  the  magnitudes  of  the  moments,  but 
their  first  proportion  as  nascent.  It  will  be  the  same  thing,  if,  instead  of 
moments,  we  use  either  tlie  velocities  of  the  increments  and  decrements 
(which  may  be  called  the  motions,  mutations,  and  fluxions  of  quantities) 
or  any  finite  quantities  proportional  to  those  velocities.  The  coefficient  of 
any  generating  side  is  the  quantity  which  arises  by  applying  the  genitum 
to  that  side. 

"  Wherefore  the  sense  of  the  lemma  is,  that  if  the  moments  of  any 
quantities  A,  K,  C,  etc.,  increasing  or  decreasing  by  a  perpetual  flux  or 
the  velocities  of  the  mutations  which  are  proportional  to  them,  be  called 
fl!,  b^  c,  etc.,  the  moment  or  mutation  of  the  generated  rectangle  AEwill 
be  aV,-\^bA  ;  the  moment  of  the  generated  content  ABC  will  be 
a^C-\- bXC-\-cA'B\    and   the   moments  of  the  generated  powers  A'-, 

A3.  A4,AiA2.  A^,  A^,  A-  ',  A-2,  A-i    will  be    2aA,   2>aK^,   ^aA^ 

^a\~^,    laA^,  i«A"^,    faA~^,   — aA-2,    — 2«A-3,    _^aA"2,    respect- 

n 

ively  :    and   in   general   that  the  moment  of  any  power  A '«     will    be 

n  —  »i 

— rtiA   '«    .    Also  that  the  moment  of  the  generated  quantitv  A^B  will  be 

2  (/AB-|-(^A2  ;  the  moment  of  the  generated  quantity  A^B^CS  will  be 
3<3:A2B4C2-(-4(5A3B3C2-|-2t\A.3B-iC  ;  and  the  moment  of  the  generated 

A3 
quantity—  or  A3B-2,   will    be    3r7A2B-2  —  2^A3B-3,   and   so   on. 

The  lemma  is  thus  demonstrated. 

"  Case  I.  Any  rectangle,  as  AB,  augmented  by  a  perpetual  flux  when 
as  yet  there  wanted  of  both  sides  A  and  B,  half  the  moments  \a  and  \b, 
was  K  —  ka  into  B  — i/^  or  K'^  —  \a^—\bK-\-^ab  ;  hut  as  soon  as  the  sides 
A  and  B  are  augmented  by  the  other  half  moments,  the  rectangle  be- 
comes A-\-^a  into  B+i^,  or  A'Q-]^hiB-\-U\-\-lnb.  From  this  rectangle 
subduct  the  former  rectangle,  and  there  remains  the  excess  «B-(-^A. 
Therefore  with  the  whole  increments  a  and  b  of  the  sides,  the  increment 
a^-\^bK  of  the  rectangle  is  generated.     Q.  E.  D." 

"  6V7>v  2.  vSupposc  AB  always  equal  to  G.  and  then  the  moment  of 
the   content  ABC  or  GC  (by  case  i)  will  be^C+rG,  that  is  (putting  AR 


38  INTRODUCTION. 

and  aB^dA  for  G  and^)  aBC-\-dAC-\-cAB.     And  the  reasoning  is  the 
same  for  contents  under  ever  so  many  sides.     Q.  E.  D." 

It  is  unnecessary  to  quote  the  demonstrations  of  the 
other  cases,  as  they  all  flow  naturally  and  logically  from  these 
which  form  the  key  to  the  whole  system. 

We  must  concede  that  this  demonstration  is  not  as  clear 
and  complete  as  could  be  desired.  Let  us,  however,  endeav- 
or to  extract  from  it  the  real,  though  perhaps  somewhat 
vague  conception  of  the  subject  which  occupied  the  mind  of 
Newton.  It  is  to  be  remarked,  however,  that  the  doctrine  of 
limits  is  nowhere  hinted  at,  but  the  results  are  direct,  posi- 
tive and  substantial. 

The  first  question  suggested  by  the  lemma  is,  what  is 
really  meant  by  the  term  "  moment."  It  might  at  first  seem 
that  the  "  moments"  of  Newton  were  in  fact  the  same  thing 
as  the  differentials  of  Leibnitz,  for  he  speaks  of  them  as 
something  (though  not  finite  quantities)  to  be  added  or  sub- 
tracted. But  a  very  little  examination  of  the  lemma  will 
dispel  the  notion.  Their  magnitudes  are  not  to  be  regard- 
ed. But  the  magnitudes  of  the  differentials  of  Leibnitz  are 
to'be  regarded  as  infinitely  small.  Again,  "  finite  particles  '" 
are  not  ''  moments,  "  but  the  ''  very  quantities  generated  by 
the  moments."  Now  the  differentials  of  Leibnitz  never 
generate  anything;  they  are  the  infinitesimal  remains  of 
increments  that  have  been  added  and  then  taken  away. 
Again,  moments  are  the  "  nascent  principles  of  finite  magni- 
tudes." But  the  "  principles  "  which  generate  "  finite  magni- 
tudes "  or  increments  can  be  nothing  else  than  the  laws 
which  control  the  changes  in  the  "  genitum  ;"  that  is,  the 
RATE  OF  CHANG r..  This  interpretation  is  confirmed  by  the 
further  statement  that  we  may  use  instead  of  them  "  the 
velocities  "  or  any  finite  quantities  [)ro[)ortional  thereto. 
Hence  we  infer  that  ^,  b^  r,which  are  called  moments,  are  in- 
tended   as  symbols  to  represent  the  rates  of  change,  being 


THE  TRUE  METHOD  OF  NEWTON.  39 

finite  quantities  proportional  to  those  rates,  and  as  the 
quantities  A,  B,  C,  etc.,  are  increasing  or  decreasing  l)y  a 
*'  perpetual  flux,  "  tliat  is  l)y  a  uniform  rate  of  change,  the 
actual  increments  or  decrements  a^  b^  c  will  represent  those 
rates.  So  that  the  difference  between  \  —  \a  and  A-f-^^ 
(equal  to  a)  represents  the  rate  of  increase  of  A,  and  the  dif- 
ference between  \\  —  \h  and  B-f-V''  (cciual  to  b)  accruiir^ 
during  the  same  time  represents  the  corresponding  rate  of 
increase  of  B  ;  and  the  ratio  of  a  to  b  represents  the  ratio  of 
those  rates  whatever  may  be  their  magnitude  as  symbols. 
But  while  these  symbols  or  suppositive  increments  (being 
produced  at  a  uniform  rate)  represent  the  res})ective  rates 
of  increase  of  A  and  B,  we  are  told  tliat  tlie  corresponding 
increment  of  their  product  (^/B+Z-A)  represents  the  ''mo- 
ment "  or  rate  of  increase  of  their  product.  Now  as  the  pro- 
duct does  not  increase  at  a  uniform  rate,  it  becomes  a 
question  why  fJiis  incrcjiiciit  should  represent  the  I'ate  of 
increase  of  AB.  This  is  probably  one  of  those  cases  in 
whicli  the  intuitive  perceptions  of  Newton  seized  the  true 
result  without  stopping  to  elaborate  the  intermediate  steps. 

The  solution  of  this  question  \\'\\\  be  found  in  this  work. 

We  have  then,  for  the  true  method  of  Newton,  the  rates 
of  change  of  the  variables,  instead  of  tlie  infinitely  small 
differences  between  their  increments.  It  is  an  interesting 
question,  why  this  method  has  not  been  more  generally  used 
by  the  teachers  of  the  Calculus. 

One  reason  is  probably,  that  although  it  was  discovered 
bv  Newton  as  early  as  1666,  it  was  not,  for  many  years,  put 
into  systematic  form  for  pul)lic  use;  not,  in  fact,  for  any 
purpose,  until  167 1,  and  then  remained  dormant,  being 
only  referred  to  in  a  lemma  in  his  Principia  published  in 
1687.  In  that  he  merely  pointed  out  the  principles,  and 
did  not  himself  give  any  thing  else  to  the  world  until,  in 
1704  he  published  a  tract   explaining   the    principles   of  his 


40  INTRODUCTION. 

method  and  applying  it  to  quadratures.  Finally,  in  17  ii,, 
Prof.  Barrow  published  a  treatise  which  Newton  had  put  into 
his  hands,  containing  a  general  method  of  calculating  the 
quadratures  of  curves,  and  also  resolving  equations.  Thus 
the  real  method  of  Newton  came  gradually  before  the  world, 
in  such  a  shape  as  not  to  attract  general  attention,  nor  to  be 
practically  used  for  making  discoveries  in  mathematics.  His 
attention  was,  no  doubt,  so  intensely  occupied  by  his  multi- 
tudinous labors  in  other  departments  of  science,  that  he 
did  not  himself,  realize  the  importance  of  his  discovery. 

Meanwhile  Leibnitz  had,  in  1684,  published  his  Differ- 
'ential  Calculus  in  which  he  gave  rules  for  applying  his 
method,  so  that  it  was  ready  for  immediate  public  use.  It 
attracted  attention  at  once,  not  merely  by  its  utility  in  the 
solution  of  difficult  problems,  but  also  by  the  new  and 
extraordinary  conceptions  it  contained  as  to  the  nature  of 
quantities,  which  conceptions  could  not  be  denied  since  the 
results  founded  on  them  were  found  to  be  true.  All  these 
circumstances  combined  to  give  to  the  method  of  Leibnitz 
a  precedence  which  it  has  continued  to  hold  from  the 
beginning. 

The  invention  of  what  is  known  as,  '"'' the 7nethod of  UiJiits^* 
had,  also,  an  influence  in  keeping  the  true  method  of 
Newton  in  the  back  ground.  This  method  is  founded  on 
the  following  lemma  in  the  first  Book  of  his  Principia. 

"  Qiiaiititics  and  ratios  of  quantities^  7vhich,  in  any  fiiiite 
time  converge  cojitinually  to  equality^  and  before  the  end  ff  that 
time  approach  nearer  the  one  to  the  other  tha?t  by  any  given 
difference^  become  ultimately  equal.'* 

From  this  circumstance,  when  it  was  formulated  and 
brought  into  use,  it  was  termed  "  Newton's  method,"  and 
thus  obtained  the  prestige  of  his  great  name.  It  was  at  once 
seized  by  those  whose  reason  was  shocked  by  the  absurdities 
of  the  infinitesimal  system,  since,  although  it  was  feeble,  it 


THE    TRUE    METHOD    OF     NEWTON.  4I 

had  no  logical  inconsistencies.  To  strengthen  it,  some 
geometers  adopted  the  symbols  of  Leibnitz,  and,  in  this  way, 
endeavored  "  to  combine  the  advantages  of  the  two  methods'"; 
and,  when  the  results  were  found  to  be  true,  were  confirmed 
in  the  use  of  this  compound  method,  without  insisting  on  a 
logical  demonstration  of  the  principle  they  assumed.  It  put 
to  rest  the  feeling  of  inconsistency,  arising  from  the  use  of 
infinitesimals,  and  they  were  content. 

Another  reason  for  the  neglect  of  Newton's  true  method, 
may  be  found  in  the  symbols  he  adopted,  which  were  not 
well  adapted  to  analytical  operations. 

"  *  The  variables  are  denoted  by  the  final  letters  of  the  alphabet;  as 
x^  y,  z  etc.  and  their  fluxions  are  indicated  by  the  same  letters  with  a  dot 
over  them.  Thus  i',  j/andi  are  the  .symbols  of  the  fluxions  o{  x,  y  and  z. 
If  the  fluxions  are  variable,  they  may  be  regarded  as  fluents,  whose  flux- 
ions may  be  taken,  and  these  are  denoted  by  the  same  letters  with  two 
or  more  dots  over  them,  according  to  the  order  of  the  fluxion.  Thus 
'y  'y  y  ctc.  dcnotc  fluxions  of  y  of  the  second,  third,  fourth  etc.  orders. 
If  the  fluent  is  a  radical,  as  ^^/x—y,  its  fluxion  is  denoted  by  placing  the 
radical  in  a  parenthesis,  and  writing  a  dot  over  it  to  the  right,  as 
(-y/x— J')'.  Also,  the  fluent  of  a  fraction  is  written  in  a  similar  man- 
ner, thus,  the  fluxion  of  —  is  written   (  —  I 

"  Sometimes  the  fluxion  is  indicated  by  the  letter  F,  and  the  fluent  by/; 
thus  J^  (V-^*"— 7)'  is  the  same  as  (V-^~J'')  • 

"Also  the  expression  of  /x\/ a—^x^  and  /  ( —j^ — )    denote  the 

fluents  of  x\^  a-\-dx^  and  — , respectively. 

'  a-\-x2        ^  ■' 

*'  This  notation  is  exceedingly  cumbrous,  particularly  in  the  higher 
branches  of  analysis,  and  for  this  reason,  principally,  the  method  of 
fluxions  has  gone  into  disuse." 

Probably,  however,   the   most   effective  reason  why   the 
true  method  of  Newton  has  been    suffered   to  die   out,  was, 

*  Davies  &  Peck  —  Mathematical  Dictionary. 


42  INTRODUCTION. 

that  its  real  essence  was  not  thoroughly  comprehended. 
Indeed  it  would  seem  that,  in  Newton's  own  mind,  it  was 
not  perfectly  clear  and  exactly  defined.  This  is  apparent  in 
the  extract  we  have  given  from  the  lemma  in  which  he  lays 
the  foundation  of  his  method.  We  have  endeavored  to 
extract  from  that  lemma  what  was  his  real,  although  some- 
what vague  idea ;  and  if  the  effort  has  been  successful,  it 
seems  to  be  quite  certain  that  most  of  his  commentators 
have  failed  to  do  so.  In  order  to  have  the  subject  fairly 
before  our  minds,  let  us  re-state  that  principle,  viz.  —  the 
function  which  Leibnitz  terms  "  diffei-ential^''  and  which 
Newton  designates  as  a  ^''  fluxion^''  is  the  concrete  symbol 
which  represents  the  rate  of  change  in  the  variable,  and 
which  consists  of  the  change  of  that  would  take  place  at  that 
rate,  continued  uniformly  for  one  unit  of  time.  It  is  not 
contended  that  this  definite  and  plain  statement  was  made 
by  Newton  ;  but  it  is  believed  that  this  was  the  true  idea 
that  was  at  the  bottom  of  his  method  of  fluxions. 

The  following  extracts  from  some  of  Newton's  commen- 
tators will  show  how  widely  their  ideas  differ  from  the  one 
we  have  just  stated.  The  following  is  from  Davies  &  Peck's 
Mathematical  Dictionary. 

"  The  idea  effluxions  and  fluents  was  first  presented  by  Newton  and 
was  based  upon  the  idea  of  motion.  According  to  his  view  a  plain  curve 
or  line  may  be  conceived  as  generated  by  a  point  moving  uniformly  in 
the  direction  of  some  fixed  line,  and  having  at  the  same  time  a  lateral 
motion  with  respect  to  this  line,  which  is  governed  by  some  law  depen- 
dent on  the  nature  of  the  curve  generally.  The  part  of  the  curve  gener- 
ated at  any  instant  of  time,  is  called  the  fluent,  and  that  infinitely  small 
element,  generated  during  the  next  infinitely  small  and  constant  period 
of  time,  is  called  its  fluxion." 

We  also  find  in  Comte's  Philosophy  of  Mathematics  the 
following  remarks  on  the  fluxional  method  of  Newton. 

"  It  is  easy,"  he  says,  "to  understand  the  general  and  necessary  iden- 
tity of  this  method  with  that  of  limits,  complicated  with  the  foreign  idea 


THE  TRUE  METHOD  OF  NEWTON.  43 

of  motion.  In  fact,  resuming  the  case  of  the  curve,  if  we  suppose,  as  we 
evidently  always  may,  that  the  motion  of  the  describing  point  is  uniform 
in  a  certain  direction,  that  of  the  abscissa,  for  example,  then  the  fluxion 
of  the  abscissa  will  be  constant,  like  the  element  of  time  ;  for  all  other 
quantities  generated,  the  motion  can  not  be  conceived  to  be  uniform 
except  for  an  infinitely  small  time.  Now  the  velocity  being,  in  general, 
according  to  its  mechanical  conception,  the  ratio  of  each  space  to  the  time 
em[)ioyed  in  traversing  it,  and  this  time  being  proportional  to  the  incre- 
ment of  the  abscissa,  it  follows  that  the  fluxions  of  the  ordinate,  of  the 
arc,  of  the  area,  etc.,  are  really  nothing  else  (rejecting  the  intermediate 
consideration  of  time)  than  the  final  ratios  of  the  increments  of  these 
different  quantities  to  the  increment  of  the  abscissa.  This  method  of 
fluxions  and  fluents  is,  then,  in  reality,  only  a  manner  of  representing  by 
a  comparison,  borrowed  from  mechanics,  the  method  of  prime  and 
ultimate  ratios,  which  alone  can  be  reduced  to  a  calculus." 

The  following  extract  from  Prof.  Loomis'  History  of  the 
Calculus  which  forms  the  introduction  to  the  last  revised 
edition  of  his  work  on  that  subject,  would  seem  to  show  that 
he  viewed  the  subject  in  the  same  light  as  Davies  and 
Comte.     He  says : 

"  Indeed  the  fluxions  and  fluents  of  Newton  correspond  essentially  to 
the  differentials  and  integrals  of  Leibnitz :  so  that  the  two  methods 
differ  only  in  the  notation,  and  in  the  peculiar  modes  of  viewing  the 
subject,  or  what  is  commonly  called  the  metaphysics  of  the  Calculus." 

In  reference  to  these  extracts  we  remark  —  First.  That 
the  idea  of  the  generation  of  a  geometrical  magnitude  by 
tne  flowing  of  an  element,  which  has  been  ascribed  to 
Newton  as  the  inventor,  and  which  it  is  said  suggested  his 
method  of  fluxions,  is  by  no  means  peculiar  to  the  calculus. 
It  is  freely  used  in  Descriptive  and  Analytical  Geometry, 
and  is  moreover,  in  many  cases,  the  best,  and  sometimes 
the  only  clear  and  concise  method,  of  defining  magnitudes  ; 
as  in  the  case  of  surfaces  and  solids  of  revolution. 

Second.  That  the  differential,  or  fluxion,  of  a  curve,  is 
"  that  infinitely  small  element  generated  during  the  next 
infinitely  small  and  constant  period  of  time"  is  simply  another 


44  INTRODUCTION. 

mode  of  stating  the  infinitesimal  theory,  which  is  certainly 
not  to  be  found  in  Newton's  explanation  of  his  own  method. 
On  the  contrary  it  is  clear  from  a  careful  examination  of  it, 
that  there  is  nothing  in  it  involving  the  idea  of  infinitesimals 
nor  of  limits.  Such  a  statement  then  could  have  proceeded 
only  from  an  entire  misapprehension  of  the  subject.  M. 
Comte  seems  to  think  that  he  is  forced  to  the  same  conclu- 
sion as  Dr.  Davies,  from  the  fact,  that  since  velocity  can  be 
measured  only  by  comparing  the  space  passed  over  at  a 
uniform  rate  with  the  time  occupied  in  traversing  it,  the 
space  and  time  must  be  infinitely  small  in  order  to  find,  at 
any  moment,  the  value  of  a  variable  velocity  ;  and  hence  it 
can  be  found,  only  by  the  method  of  limits.  If  his  premises 
were  true,  his  conclusion  would  be  reasonable.  But  it  is  not 
true  that  variable  velocity  is  so  measured ;  and  it  is  surpris- 
ing that  a  philosopher  should  forget  that  measure  of  velocity 
which  is  used  by  the  whole  world,  every  day,  in  common 
conversation.  Namely,  the  space  that  ivoiUd  be  uiiiformly passed 
over  in  any  unit  of  time — at  the  same  rate;  as  when  we  say 
a  body  is  falling  to  the  earth  with  a  velocity  of  fifty  feet  in 
one  second  of  time.  Such  a  statement  could  have  been 
made,  only  for  the  purpose  of  sustaining  a  pre-conceived 
theory  founded  on  an  entire  misconception  of  Newton's 
method.  These  samples  of  this  misconception  are  taken 
from  modern  writers,  but  they  have  only  followed  the 
notions  that  have  been  current  in  the  mathematical  world 
from  the  beginning,  and  prove  that  up  to  their  time  the 
principle  of  Newton's  method  had  not  been  apprehended. 

I  have  thus  endeavored  to  show  that  the  true  method  of 
Newton  has  been  misunderstood,  but  if,  in  fact,  the  writer 
has  mistaken  the  method  of  Newton,  and  his  commentators 
have  comprehended  it  aright,  then  the  following  work  com- 
prises A  NEW  METHOD,  and  Can  stand  on  its  own  merits 
without  the  support  of  Newton's  great  name.    The  reader  is. 


THE    TRUE    METHOD    OF     NEWTON.  45 

at  liberty  to  form  his  own  opinion  as  to  the  paternity  of  the 
"-  direct  method  of  rates  ;''  \i  \\Q\'i  unwilling  to  ascribe  it  to 
the  great  genius  of  the  seventeenth  century,  the  writer  will 
cheerfully  stand  "  in  loco  parentis." 


PART    I, 


Differential  Calculus. 


Differential  Calculus. 


SECTION  I. 


DEFINITIONS  AND  FIRST  PRINCIPLES. 


VARIABLES. 

(I)  Two  classes  of  quantities  are  considered  in  the  dif- 
ferential calculus,  namely,  vai'iablcs  and  constants.. 

Variables  are  quantities  that  are  in  a  state  of  cha?ige ;  that 
is,  their  values  are  in  an  increasing  or  decreasing  condition ; 
such,  for  example,  as  the  quantity  of  water  in  a  vessel  which 
is  being  filled  or  emptied  by  a  continuous  stream ;  or  as  the 
force  of  attraction  which  increases  or  diminishes  as  the 
attracting  bodies  approach  or  recede  from  each  other;  or  as 
the  space  between  these  same  bodies  while  they  are  moving. 
They  are,  '\\\  the  differential  calculus,  not  merely  quantities 
subject  to  change,  or  to  which  different  values  may  be  as- 
signed ;  but  quantities  in  which  the  change  is  supposed  to  be 
actually  occurring  at  the  moment  when  they  become  the  sub- 
ject of  the  analysis.  It  is  their  actual  co7uiition  and  not  their 
attributes  or  qualities  that  are  referred  to  in  this  definition. 
Take  for  example  the  space  passed  over  by  a  falling  body. 
That  space  is  a  variable,  not  because  it  may  or  does  have 
different  values,  but  because  its  value  is  constantly  cha?igingy 

4 


50  DIFFERENTIAL    CALCULUS. 

or  is  in  a  state  of  chmige.  It  is  this  state^  and  not  any  actual 
change,  that  is  the  peculiar  subject  of  the  transcendental 
analysis. 

RATE    OF    VARIATION. 

(2)  Rate  of  variation  is  the  relation  between  the  change 
of  a  variable  and  the  tiine  occupied  by  the  change.  Being  a 
relation  and  not  a  simple  quantity,  it  can  only  be  represented 
by  a  symbol^  which  is  a  uniform  cha?ige  in  a  given  unit  of  time. 
If  the  rate  is  constant,  then  the  actual  change  is  the  true 
symbol,  but  if  it  is  variable,  then  the  change  must  be  a  sup- 
positive  one  —  that  is,  one  that  luould  take  place  in  the  same  tinit 
of  time  if  it  were  to  continue  uniform.  Thus  in  the  case  of  a 
falling  body,  it  is  said  the  velocity  at  a  certain  moment  is  so 
many  feet  in  one  second.  It  is  not  meant  that  the  body  ac- 
tually falls  through  that  distance  in  one  second,  nor  any  dis- 
tance whatever  at  that  rate,  but  that  it  would  fall  so  far  if  the 
velocity  existing  at  that  moment  were  to  continue  uniform 
for  one  second.  The  velocity  belongs  to  that  one  moment, 
and  that  one  position  only.  At  the  very  next  point  above 
and  below  this  position  the  velocity  is  different,  and  hence  no 
actual  movement,  however  small  in  respect  to  space  and  time, 
can  possibly  represent  it.  This  will  be  seen  at  once  if  we 
consider  what  velocity  is.  It  is  not  of  itself  a  quantity,  but 
a  relation,  which  refers,  not  to  the  place,  but  to  the  condition 
of  the  body  in  respect  to  the  motion  —  that  is,  to  the  degree 
or  intensity  of  the  state  of  motion  in  which  the  body  is. 

So  it  is  with  all  variables.  The  rate  of  change  refers  to  the 
intensity  with  which  the  change  is  going  on,  and  if  it  is  not 
uniform  it  can  not  possibly  represent  the  rate,  for  it  lacks  the 
essential  element  of  the  required  symbol.  The  latter  must 
therefore  be  obtained  from  the  law  whicli  governs  the  change 
and  not  from  the  change  itself.  Hence  instead  of  giving  to 
a  function  an  actual  increment  for  the  sake  of  obtaining  its 


DEFINITIONS    AND     FIRST    PRINCIPLES-  5 1 

rate  of  increase  at  any  moment,  we  examine  the  law  which 
governs  the  change ;  and  the  expression  of  this  law  is  the 
change  that  7vould  take  place  in  a  unit  of  time  if  the  rate  were  to 
continue  uniforni  ;   and  this  is  the  measure  of  the  hate. 

Note. —  It  must  be  remarked  that  tlie  ideas  of  time,  motion  and  velocity,  attached 
to  the  ordinary  meaning  of  these  words,  have  no  place  in  the  abstract  science  of  the 
differential  calculus.  The  terms  motion  and  velocity  are  used  in  tins  article  merely  to 
illustrate  the  meaning  of  the  term  "ra/^."  It  is  true  that  velocity  is  a  rate — the 
rate  of  motion.  But  many  other  things  beside  motion  have  a  rate  ;  such  as  the  varia- 
tion of  light,  heat,  magnetism,  force,  anything  which  increases  or  diminishes  by  the 
operation  of  prescribed  law  ;  and  the  calculus  is  applicable  to  all  such  subjects  where 
the  conditions  can  be  expressed  analytically. 

The  idea  of  time  in  its  absolute  sense  is  also  foreign  to  the  calculus.  The  term 
"  unit  of  time  "  in  the  definition  does  not  refer  to  any  specific  portion  of  time  ;  it  may 
be  great  or  small ;  its  value  does  not  enter  into  the  calculation,  and  hence  this  system 
does  not  in  any  wise  invade  the  domain  of  natural  philosophy.  All  that  the  abstract 
science  of  the  calculus  has  to  do  with  tiuie^  is  confined  to  the  simple  condition  that  the 
suppositive  changes  in  the  value  of  the  variable  and  of  its  function,  which  symbolize 
their  rates  of  change,  shall  be  simziltaneoiis.  And  this  is  no  more  than  all  systems  of 
the  calculus  require  for  the  actual  cha.x\gQS  which  are  supposed  to  be  made  in  the  same 
quantities. 

In  the  application  of  the  calculus  to  Geometry  the  idea  of  motion  was  in  some  sort 
introduced  ;  but  not,  however,  in  its  philosophical  sense  as  having  an  absolute  value. 
Geometrical  magnitudes  are  supposed  to  be  generated  by  the  movement  of  their  ele- 
ments. Thus  a  line  is  generated  by  the  flowing  of  a  point,  a  surface  by  a  line,  and  a 
solid  by  a  surface  ;  and  this  conception  is  used  to  determine  the  proportion  of  magni- 
tudes, by  comparing  the  rates  at  which  they  are  generated  instead  of  comparing  the 
magnitudes  themselves  with  each  other.  This  idea  of  the  generation  of  magnitudes 
by  means  of  their  elements  is  not  new  in  mathematics.  It  is  one  of  the  seminal  ideas 
of  the  Cartesian  system  ;  and  though  in  this  work  it  is  certainly  made  more  promi- 
nent than  it  has  usually  been,  and  more  prolific  in  results,  it  is  not  therefore  out  of 
place. 

DIFFERENTIALS. 

(3)  The  differential  of  a  variable^  or  fu?iction^  is  its  rate  of 
change  or  variation^  symbolically  expressed  by  the  suppositive 
change  that  would  take  place  at  that  rate. 

If  the  variable  is  essentially  positive  and  increasing^  or  neg- 
ative and  decreasing^  its  differential  will  be  essentially  posi- 
tive. If  it  is  essentially  negative  and  increasing.^  or  positive 
and  decreasing^  its  differential  will  be  essentially  negative. 
Thus,  if   we  consider  a   northern    latitude    positive   and   a. 


52  DIFFERENTIAL    CALCULUS. 

southern  negative,  a  vessel  will  have  a  positive  rate  (or  dif- 
ferential) of  progress  if  her  northern  latitude  is  increasing  or 
her  southern  latitude  is  decreasing;  and  vice  versa  her  rate 
■of  progress  will  be  negative. 

The  notation  used  to  designate  the  rate  of  variation  or  dif- 
ferential is  the  letter  d  placed  before  the  variable  whose 
rate  is  required.  Thus  the  differential  of  x  is  written  dx. 
If  the  variable  is  a  component  expression  such  as  x^-\-ay^ 
the  differential  would  be  written  d{x^-\-ay).  Variables  are 
themselves  indicated  by  the  last  letters  of  the  alphabet. 

Note. —  I  use  the  nomenclature  and  notation  of  Leibnitz,  not  because  there  is  any 
actual  necessity  for  so  doing,  but  because  their  use  has  become  so  general,  not  only  in 
the  system  of  Leibnitz,  but  also  in  other  systems,  that  it  seems  to  have  become  fixed, 
■without  much  regard  for  their  original  derivation  and  meaning  ;  and  hence  a  change 
:in  that  respect  would  appear  like  an  unnecessary  innovation. 

The  term  "  fluxion  "  is  more  truly  significant  of  the  true  principles  of  the  science 
than  the  term  "  differential,"  and  the  symbol  "  d  "  is  no  better  than  some  other  would 
be  :  but  it  is  just  as  good  as  any,  and  the  use  of  it  involves  no  inconsistency  with  the 
principles  on  which  the  system  is  based, 

CONSTANTS. 

(4)  The  other  class  of  quantities  which  enter  into  the 
transcendental  analysis  is  that  of  constants.  These  are  sup- 
posed to  have  a  fixed  value,  although  it  is  not  always  neces- 
sary that  this  value  should  be  known  or  given.  In  equa- 
tions the  constant  quantities  express  the  conditions  of  the 
proposition,  and  while  they  are  generally  supposed  to  have 
a  given,  or,  at  least,  an  assignable  value,  there  are  many  cases 
in  which  their  value  must  be  determined  by  the  solution  of 
an  equation  just  as  any  unknown  quantity  in  algebra  is  de- 
termined. This,  however,  does  not  make  them  variables  ; 
their  value  is  as  much  fixed  as  if  it  were  known  at  first.  The 
solution  of  an  equation  is  rendered  necessary  in  order  to 
make  the  immediate  conditions  conform  to  some  ulterior 
conditions  imposed  upon  them.  Thus  the  general  equations 
of  two  circles  will  determine  the  curves  when  the  constants 


DEFINITIONS    AND    FIRST     PRINCIPLES.  53 

are  given  ;  but  if  there  is  an  ulterior  condition  that  they 
must  be  tangent  to  each  other,  the  constants  must  be  made 
to  conform  to  this  condition ;  which  can  be  done  only  by  an 
equation  from  which  the  necessary  values  can  be  obtained. 
This  solution  does  not  fix  the  values  of  the  constants,  but 
only  makes  known  those  values  which  were  fixed  or  rendered 
certain  by  the  conditions  to  which  they  were  subjected.  A 
constant  then  is  never  in  a  state  of  variation. 

FUNCTIONS. 

(5)  A  function  of  a  variable  is  any  algebraic  expression  whose 
value  depends  on  that  of  the  variable.     Thus 

ax'  —  2x 

is  a  function  of  x  since  its  value  changes  with  that  of  x^ 
supposing  a  to  be  constant.     The  expression 

ax-\-by 

is  a  function  of  x  and  y  {a  and  b  being  constant),  for  it  de- 
pends on  both  X  and y  for  its  value;  and  thus  we  may  have 
a  function  of  any  number  of  variables. 

When  the  expression  does  not  involve  an  equation.,  the 
variables  are  independent  of  each  other;  that  is,  we  may 
assign  to  any  of  them  any  value  whatever  without  regard  to 
the  values  assigned  to  the  rest.  But  an  equation  which  con- 
tains variables  will  have  at  least  o?ie  dependent  on  the  others 
for  its  value.     Thus  in  the  equation 

y'-\-bu^zax 

in  which  x,  y  and  ic  are  variables,  we  may  give  arbitrary 
values  to  any  two  of  them,  but  the  value  of  the  third  must 
be  determined  from  the  equation.  This  last  is  called  a  func- 
tion of  the  others  and  the  dependent  variable,  while  the  oth- 
ers are  called  independent  variables.  When  the  dependent 
variable  stands  alone  in  one  member  of  the  equation   it  is 


>' 


54  DIFFERENTIAL    CALCULUS. 

called  an  explicit  function  of  the  others,  but  when  combined 
with  the  others  it  is  called  an  implicit  function.  The  term 
function,  however,  applied  to  the  dependent  variable  is  to 
be  understood  as  meaning  the  representative  of  the  function, 
<and  not  literally  the  function  itself. 

Functions  are  commonly  divided  into  two  classes,  which 
are  distinguished  by  the  manner  in  which  the  variables  enter 
into  them,  and  are  called  ^''Algebraic''''  and  ^''Transcendental. 

Algebraic  functions  are  those  in  which  the  variables  are  sub- 
jected only  to  the  operations  of  addition^  subtraction^  multiplica- 
tion^ division.,  and  involution  or  evolution,  denoted  by  constant  ex- 
ponents or  indices. 

Trausce/idental  functions  are  those  in  which  the  variable  is 
either  an  exponent,  logarithm  or  trigonometrical  line.,  such  as  a 
sine.,  tangent,  etc. 

(6)  The  fundamental  problem  of  the  calculus  is  to  find 
the  differential,  or  rate  of  change,  in  a  function  of  a  variable 
produced  by  that  of  the  variable  itself. 

As  the  differential  or  rate  of  change  in  a  variable  is  rep- 
resented by  a  suppositive  change,  taking  place  at  a  uniform 
rate,  and  that  of  the  function  arising  from  it  by  a  correspond- 
ing suppositive  change,  these  changes  (being  uniform  from 
the  beginning)  will  have  a  constant  ratio  independent  of  their 
value.  Hence  the  differential  of  the  variable  is  always  a  factor 
of  the  differential  of  its  function.  The  other  factor,  that  is,  the 
ratio  between  the  differential  of  the  variable  and  that  of  its 
function,  is  called  the  differential  coefficient  of  the  function. 
Since  this  ratio  is  not  affected  by  the  value  of  the  suppositive 
change  representing  the  differential  of  the  variable,  this  dif- 
ferential is  indicated  ])y  an  indeterminate  symbol,  and  the 
differential  of  the  function  becomes  a  function  of  that  symbol. 

The  differential  of  a  function  is  o])tained  by  a  process 
called  differentiation^  and  the  differential  coefficient  is  ob- 
tained by  dividing  the  differential  of  the  function  by  that 


DEFINITIONS    AND    FIRST    PRINCIPLES.  55 

of  the  variable;  so  that  the  differential  coefficient  of  ax'—bx 

would  be 

d{^ax^  —  bx^ 

dx 
If  we  represent  the  function  by  ti  we  have 

u^^ax'  —  hx 
and 

dtt 

T-  =  differential  coefficient,  which 

we  can  obtain  as  soon  as  we  know  how  to  find  the  differen- 
tial of  ax^—bx. 

(7)  If  we  have  an  equation  containing  variables  in  each 
member,  since  the  two  members  are  always  equal,  their  rates 
of  change  are  also  equal;  hence  we  may  differentiate  each 
member  as  a  separate  function,  and  place  the  results  equal 
to  each  other.  If  the  equation  contains  more  than  one  vari- 
able, one  of  them  will  be  dependent  and  the  value  of  its 
differential  will  depend  on  the  values  of  the  other  variables 
and  their  differentials.  Either  of  the  variables  may  be 
taken  as  the  dependent  one,  and  it  will  then  represent  a 
function  of  the  rest. 

If  the  differential  of  a  function  of  two  or  more  variables 
be  taken  with  reference  to  07ie  only,  and  then  divided  by  Us 
differential,  the  result  will  be  the  differential  coefficient  for 
that  variable,  and  all  the  rest  must  be  treated  as  constants 
for  that  coefficient^  and  the  function  as  a  function  of  that  va- 
riable; for  a  differential  coefficient  can  exist  only  between  a  single 
variable  and  its  functiofi. 

If  we  wish  to  indicate  a  function  of  any  variable,  as  a% 
vvithout  giving  it  any  particular  form,  for  the  purpose  of 
demonstrating  some  general  truth  applicable  to  all  forms, 
we  use  the  expression  F  (.t),  which  means  any  function  de- 
pending on  X  for  its  value.  If  it  is  a  function  of  two  or 
more  variables,  the  expression  is  F  (.v,  >'),  or  F  {x^y^  s),  and 
similarly  for  a  greater  number  of  variables. 


SECTION    II. 


DIFFERENTIA  TION  OF  FUNCTIONS. 

Proposition  I. 

(8)  To  find  the  sign  with  which  the  differential  of  the  varia- 
ble must  enter  that  of.  the  fimction  to  which  it  belongs. 

Among  the  characteristics  of  quantity  as  used  in  the  cal- 
culus, is  that  of  being  essentially  positive  or  negative,  ac- 
cording as  it  lies  on  one  side  or  the  other  of  the  zero  point. 
If,  for  instance,  the  value  is  reckoned  toward  the  negative 
side  it  is  essentially  negative  independent  of  the  algebraic 
sign  that  may  be  prefixed  to  it.  Thus  the  co-sine  of  120°  is 
essentially  negative,  whether  it  is  to  be  added  or  subtracted,, 
while  the  sine  of  the  same  angle  is  essentially  positive. 
These  characteristics  are  independent  of,  and  wholly  dis-^ 
tinct  from,  the  algebraic  signs  that  may  be  prefixed  to  them. 
Hence  when  quantities  enter  into  a  function  as  variables  we 
must  inquire  what  will  be  the  effect  of  these  characteristics 
on  the  influence  which  their  rate  of  change  will  have  on 
that  of  their  function,  so  that  we  may  give  to  the  differential 
of  the  variable  that  sign  which  will  produce  a  rate  of  change 
in  the  function  in  the  right  direction. 

If,  then,  a  variable,  whether  intrinsically  positive  or  neg- 
ative, is  increasing,  its  rate  of  change  will  affect  the  rate  of 
its  function  in  the  same  direction  as  its  value  affects  the  value 

56 


DIFFERENTIATION    OF    FUNCTIONS.  57 

of   tlie  function;  and,  hence,  having  essentially    the   same 
character  (Art.  3)  it  must  have  the  same  sign. 

If  it  is  diminishing,  its  rate  of  change  will  affect  that  of 
its  function  in  a  direction  contrary  to  that  in  which  the  7'aliie 
of  the  variable  affects  the  ralue  of  its  function;  but  having 
itself  a  character  contrary  to  that  of  the  variable  (Art.  3)  it 
must  still  have  the  same  sign,  in  order  to  produce  the  proper 
effect. 

Let  us  for  instance  take  the  function 

X — y 
in  which  x  is  a  positive  and  increasing  variable.  Now  if  y 
is  increasing  its  rate  will  be  essentially  positive  or  negative, 
according  as  y  is  itself  essentially  positive  or  negative  (Art. 
3),  and  must  therefore  affect  the  differential  or  rate  of  the 
function  (to  increase  or  diminish  it)  in  the  same  direction  as 
y  affects  its  value.  \i y  is  diminishing,  its  rate  of  change  will 
be  essentially  negative  '\i  y  is  positive,  and  positive  if  )'  is 
negative  (Art.  3),  and  will  therefore  affect  the  rate  of  the 
function  in  a  direction  contrary  to  that  in  which  y  affects  its 
value ;  but  being  essentially  contrary  in  its  nature  to  that  of 
)",  it  must  enter  the  differential  of  the  function  with  the 
same  sign  in  order  to  produce  a  contrary  effect. 

Thus,  whether  the  variable  be  intrinsically  positive  or 
negative,  whether  it  is  increasing  or  diminishing,  the  sign 
prefixed  to  its  differential  must  in  all  cases  be  the  same  as  that 
prefixed  to  the  variable  itself. 

ILLUSTRATION. 

If  we  consider  a  northern  latitude  })ositive  and  a  southern 
one  negative,  and  there  are  two  vessels,  A  and  B,  sailing 
north  of  the  equator,  let  the  latitude  of  A  be  represented  by 
X  and  that  of  B  by  y.  Then  the  difference  of  their  latitudes 
will  be  .V— _,r.  If  both  are  sailing  north  their  rates  of  prog- 
ress will  be  positive,  and  the  rate  of  change  in  their  differ- 


58  DIFFERENTIAL    CALCULUS. 

ence  of  latitude  will  be  the  difference  of  their  rates  of  sail- 
ing; hence  the  rate  of  B,  which  is  the  differential  oi  y,  wiJl 
have  a  minus  sign.  If  B  is  sailing  south,  the  difference  of 
their  latitudes  will  be  still  x—y;  but  since  the  rate  of  change 
in  this  difference  is  the  real  sum  of  their  rates  of  sailing,  the 
rate  of  B  being  essentially  negative,  (x\rt.  3)  must  have  a 
minus  sign,  so  that  the  algebraic  difference  will  produce  the 
real  sum. 

If  B  be  south  of  the  equator,  the  difference  of  their  lati- 
tudes will  be  their  real  sum,  butj  being  now  essentially  neg- 
ative must  have  a  minus  sign  to  produce  this  sum,  and  it 
will  still  be  expressed  by  x—y.  If  B  be  sailing  south  (A 
being  still  sailing  north),  the  rate  of  B  will  be  negative  (Art. 
3),  and  since  the  rate  of  change  in  the  difference  of  their 
latitudes  is  the  real  sum  of  their  rates  of  sailing,  the  rate  of 
B  must  have  a  minus  sign,  so  that  the  algebraic  difference 
will  be  the  real  sum.  If  B  be  sailing  north,  its  rate  will  be 
positive  (Art.  3),  and  since  in  this  case  the  rate  of  change 
in  the  difference  of  latitudes  will  be  the  real  difference  of 
their  rates  of  sailing,  the  rate  of  B  must  still  have  a  minus 
sign,  so  that  the  algebraic  difference  will  correspond  to  the 
real  difference.  Hence,  if  y  enter  the  function  with  a  minus 
sign,  its  differential  or  rate  of  change  will  have  a  minus 
sign,  whether  y  is  intrinsically  positive  or  negative,  or  is  in- 
creasing or  diminishing.  A  similar  result  would  follow  if 
the  sign  wqyq  p/us^   the  sign  of  the  differential  would  be  plus. 

Proposition   II. 

(9)  To  find  the  diff ere  )2tial  of  a  function  consisting  of  terms 
connected  together  by  the  signs  plus  and  minus. 

That  is  to  say,  to  find  the  rate  of  change  in  the  function 
arising  from  the  rates  of  the  variables  which  enter  into  it. 

Every  term  in  an  algebraic  expression  may  be  considered 


DIFFERENTIATION    OF    FUNCTIONS.  59 

as  having  a  single  value,  made  up,  of  course,  of  the  respec- 
tive values  of  the  quantities  that  compose  it,  and  their  rela- 
tions to  each  other,  and  may,  therefore,  be  expressed  by  a 
single  letter.  If  a  term  contain  none  but  constant  quanti- 
ties the  letter  representing  it  will  be  considered  as  a  con- 
stant. If  it  contain  variables,  the  letter  representing  it  will 
be  considered  as  a  variable  having  the  same  rate  of  change 
as  would  arise  in  the  term  itself  from  the  rates  of  change  in 
the  variables  which  enter  into  it. 

It  will,  therefore,  be  sufficient  to  investigate  the  case  of  a 
function  in  which  each  one  of  the  terms  is  represented  by  a 
single  letter. 

Let  us  suppose  some  of  these  terms  to  be  variable  and 
others  constant,  and  the  varial)les  to  be  changing  their  val- 
ues at  any  rate  whatever,  either  uniform  or  variable,  and 
each  one  independent  of  the  rest  The  constants  will,  of 
course,  have  no  rate  of  change,  and  will,  therefore,  not  affect 
the  rate  of  change  in  the  function. 

The  differential  of  each  variable  will  l)e  the  suppositive 
uniform  change  that  ivou Id  take  place  in  it  in  a  unit  of  time 
at  the  rate  existing  at  the  moment  of  differentiation;  and 
the  differential  of  the  function  is  the  uniform  change  that 
would  take  place  in  it  arising  from  the  supposed  uniform 
changes  in  the  variables.  Now,  if  we  suppose  this  symbolic 
or  suppositive  change  to  be  made  in  each  variable,  the  cor- 
responding change  in  the  function  will  be  the  algebraic  sum 
of  the  changes  in  the  variables;  and  as  these  are  by  su])po- 
sition  uniform,  the  change  in  the  function  will  be  uniform 
also  at  the  rate  at  which  it  commenced,  and  will,  tlievefore, 
be  the  symbol  of  tliat  rate  or  the  differential  of  the  function. 

For  example,  let  us  take  the  function 
x—y-\-a—b-\-z-\-c 
in   which  x,  y  and  z  are  variables,  and   a  b  and  c  are  con- 
stants, and    rejH-esent    by   dx,  dy  and  dz  the   differentials   or 


6o  DIFFERENTIAL    CALCULUS. 

uniform  changes  that  would  take  place  in  x,  y  and  s  in  a 
unit  of  time  from  the  moment  of  differentiation.  Let  us 
also  suppose  these  symbolic  changes  to  take  place;  the  func- 
tion would  then  become 

x-{-(ix—y — c/y  -\-  a  —  b-\-  z  -^  dz-\-  c 
(Art.  8)  and  if  from  this  we  subtract  the  primitive  function 

we  have 

(fx  —  dy-\-dz 

Which  represents  the  uniform  change  in  the  function  aris- 
ing during  the  same  unit  of  time  from  the  suppositive 
uniform  changes  in  the  variables.  It  is,  therefore,  the 
symbol  representing  the  corresponding  rate  of  change,  or 
differential  of  the  function.     Hence 

d{^x—y-\-a  —  b-\-z-\-c)  =  dx—dy-\-dz 
or  the  differential  of  a  function  composed  of  terjns  containing  in- 
dependent variables^  having  any  rates  of  change  ivhatever^  the 
terms  being  connected  together  by  the  signs  plus  and  viimis^  is  the 
algebraic  sum  of  the  differentials  of  the  terms  taken  separately 
with  the  same  signs. 

Since  each  of  the  terms  in  the  case  given  may  represent 
a  compound  term  of  any  form  whatever,  it  is  now  necessary 
to  examine  the  method  of  finding  the  differential  of  a  single 
term  in  every  form  in  which  it  may  occur. 

The  number  of  these  forms  for  algebraic  terms  is  limited 
to  seven,  as  follows  : 

1.  A  variable  multiplied  by  a  constant. 

2.  One  variable  multiplied  by  another. 

3.  A  variable  divided  by  a  constant. 

4.  A  constant  divided  by  a  variable. 

5.  One  variable  divided  by  another. 

6.  A  power  of  a  variable. 

7.  A  root  of  a  variable. 

These  simple  forms,  or  some  combinations  of  them,  which 
can  be  dissected  and  operated  by  the  same  rules,  constitute 
all  that  can  l)e  assumed  by  single  algebraic  terms. 


DIFFERENTIATION    OF    FUNCTIONS.  6 1 

(10)    To  find  the  differential  of  a  variable  multiplied  by  a 

constant  quantity. 

We  have  seen  that 

d{^x-\-y-\-zA;-ii)  =  dx-\-dy-\-dz-\-du 

If  we  make  these  variables  each  equal  to  .r,  we  shall  have 

X  -\-y  -\-z-\-u=/[x 

and 

dx-\-dv-\-d3-i-dn  =  ^dx 

hence 

d{4x)^4dx 

As  the  same  reasoning  will  extend  to  any  number  of  terms, 

we  may  make  the  equation  general,  and  we  have 

d{nx)  =  ndx 

That  is,  the  rate  of  change  of  n  times  x  is  equal  to  n  times 
the  rate  of  change  of  x. 

Hence,  t/ie  differential  of  a  variable.,  7uith  a  constant  coeffi- 
cient, is  equal  to  the  differential  of  the  variable  ?nultiplied  by  the 
coefficient.  In  other  words,  the  coefficient  of  the  variable 
will  also  be  the  coefficient  of  its  differential, 

EXAMPLES. 

Ex.     I.  What  is  the  differential  of  abz"^  —  Ans.  abdz. 

Ex.     2.  What  is  the  differential  of  by"^. — Ans.  b'dy. 

Ex.     3.  What  is  the  differential  of  ax-\^c\''^  —  Ans.  adx-\-cdy. 

Ex.     4.  What  is  the  differential  of  x  —  by'?  —  A)is. 

Ex.     5.  What  is  the  differential  of  {a-^b)x} — Afis. 

Ex.     6.  What  is  the  differential  of  (e—d)y}  —  Ans. 

Ex.     7.  What  is  the  differential  of  ax-\-by-\-cz?  —  Ans. 

Ex.     8.  What  is  the  differential  of  b'^u-j-c'z? — Ans. 

Ex.     9.  What  is  the  differential  of  cvbx-j-c'dy?  —  A;is. 

Ex.  10.  What  is  the  differential  of  a'y  —  b'x?  — Ans. 

Ex.  II.  What  is  the  differential  of  b{ay—cx)?  —  A;is. 

Ex.  12.  What  is  the  differential  of  c'{bx-\-az)?  —  Ans. 


62  DIFFERENTIAL    CALCULUS. 


LEMMA. 


(II)  If  two  variables  are  increasing  at  a  uniform  rate^ 
their  rectangle  will  be  increasing  at  an  accelerated  rate,  but 
the  acceleration  will  be  constant. 

Let  X  and  y  be  increasing  uniformly  at  rates  represented 
by  dx  and  dy.  Then  dx  and  dy  will  be  the  actual  increments 
of  X  and  y  m  ^  unit  of  time,  and  at  the  end  of  in  such  units 
xy  will  have  become 

{x-\-j?idx)  {y-\-mdy)  =  xy  +  mydx -{- mxdy-\- /u'dxdy         ( i ) 

In  one  more  unit  of  time  we  shall  have 
{x -f- {m-{- 1 )dx)  ( y-\-(m-{- 1 )dy) ^=xy-\-{7/i-\- i)ydx  +  {m-\- 1 )xdy 
-{-{m-\-i  ydxdy  (  2  ) 

In  still  another  unit  of  time  we  shall  have 
{x-{-{7/i-\-2)dx)  {y-\-{?n-{-2)dy) ^xy-\-{>n-\-2)ydx-\-{in-\-2)xdy 
■\-{in-\-2ydxdy  (3) 

Subtracting  the  second  member  of  (i)  from  that  of  (2)  we 

have 

ydx + xdy  +  (  2  w  + 1  )dxdy  (4) 

and  subtracting  the  second  member  of  (2)  from  that  of  (3) 
we  have 

ydx -\- xdy  -j-  (  2 ;;/  +  3  )^/^<'/k  (5  ) 

and  subtracting  (4)  from  (5)  we  have 

2dxdy  (6) 

Now  the  expression  (4)  is  the  increment  of  the  product 
arising  from  the  uniform  increments  of  the  variable  factors 
during  one  unit  of  time,  and  expression  (5)  is  the  increment 
of  the  product  during  the  next  equal  unit  of  time  arising  from 
the  next  e(pial  uniform  increments  of  the  variables.  These 
increments  of  the  product  may  therefore  be  taken  to  represent 
its  successive  mean  rates  of  increase  arising fi^m  the  uniform 
increase  of  the  variable  factors  during  two  ecpial  successive 
units  of  time;  an^  the  exi)ression  (6),  which  is  the  differ- 
ence between  these  rates  will  represent  their  acceleration. 


DIFFERENTIATION    OF    FUNCTIONS.  6^ 

Now  since  this  last  is  a  constant  quantity  and  independent 
of  m,  it  follows  that  the  acceleration  of  the  mean  rates  of 
increase  of  the  product  will  be  constantly  the  same  during 
every  two  consecutive  units  of  time,  while  the  factors  are 
increasing  at  a  uniform  rate.  And  since  the  increase  of  the 
variables  is  continuous  and  uniform,  that  of  the  product  will 
also  be  continuous  and  according  to  a  uniform  law  of  some 
kind ;  and  since  for  every  possible  variation  in  the  number 
and  value  of  the  units  of  time,  and  in  the  value  of  the  rates 
t/x  and  i/y  the  acceleration  of  the  rate  of  increase  of  the 
product  is  constant  for  successive  periods,  it  must  be  so 
continuously,  and  equal  to  twice  the  product  of  the  rates  of 
increase  of  the  variable  factors. 

Proposition  IV. 

(12)    To  find  the  differ e7itial  of  the  product  of  two  mdepe?ident 
variables. 

Let  us  suppose  the  two  variables  A  and  B  to  be  increasing 
at  any  rate  whatever,  either  uniform  or  variable,  and  inde- 
pendent of  each  other.  Suppose  also  that  when  A  has  be- 
come equal  to  x,  B  will  have  become  equal  to  y^  and  that  dx 
and  dy  represent  their  respective  rates  of  increase  at  that 
instant;  then  they  will  represent  the  uniform  increments 
that  would  be  made  by  A  and  B  respectively,  in  the  same  unit 
of  time,  at  these  rates;  and  these  suppositive  increments 
are  what  we  have  to  consider.  Suppose  again  that  one-half 
of  each  increment  be  made  imuiediately  before  A  and  B  be- 
come equal  to  X  and  y^  and  the  other  half  afterwards.  In 
the  first  case  the  product  of  A  and  B,  or  AB,  at  the  begin- 
ning of  the  increment  will  be  equal  to 

(.T—  y^.dx')  (j'—  V2.dy^  =  xy—  %ydx—  %xdy-\-]^dxdy 
and  at  the  end  of  the  unit  of  time  it  will  be  equal  to 

(x-{-j4dx)  {y+y2dy)  =  xy-i-^Aydxi-%xdy-{-j{dxdy 
Subtracting  the  first  oroduct  from  the  last  we  have 

ydx-}-xdy 


64  DIFFERENTIAL    CALCULUS. 

which  represents  the  difference  between  the  two  states  of 
the  rectangle  AB,  or  the  increment  made  by  it,  while  the 
factors  are  passing  from  x—}^i/x  and  y—%^y  to  x-{-}4^^ 
and  y-i-%^tVj'  that  is,  while  the  variables  are  receiving  the 
uniform  increments  represented  by  i/x  and  ^iy,  their  respec- 
tive rates  of  increase  at  the  instant  they  are  equal  to  x  and 
y,  the  rectangle  is  receiving  an  increment  represented  by 
yi/x-{-x{/y.  We  are  now  to  show  that  this  increment  rep- 
resents the  rate  of  increase  of  AB  at  the  moment  that  ^x 
and  dy  represent  the  rates  of  increase  of  A  and  B  sepa- 
rately, namely,  at  the  instant  they  become  equal  to  x  and  y. 
This  suppositive  increment  would  not,  of  course,  be  made 
at  a  uniform  rate,  but  as  we  have  seen  (lemma)  at  a  um- 
formly  increasing  rate.  Hence  when  AB  would  become  xy, 
and  the  variables  had  received  half  their  suppositive  incre- 
ments, the  increment  of  AB  would  have  received  half  the 
increase  of  its  rate,  which  would  then  have  become  equal  to 
its  mean  rate  for  that  unit  of  time.  But  the  mean  rate  is 
that  by  which  the  xwQ^XQm.o.wX.woulabe  made  in  the  same  time 
if  it  were  uniform,  and  if  the  increment  were  made  at  a 
uniform  rate  it  would  measure  the  rate  of  increase  of  the 
rectangle.  Now  the  actual  increment  (represented  by  ydx 
-\-xdy)  of  the  rectangle,  being  made  in  the  same  time,  as  it 
would  be  if  made  uniformly  at  its  mean  rate,  existing  when 
A  and  B  were  equal  to  x  and  y,  is  the  true  measure  of  that 
rate;  that  is,  of  the  rate  of  increase  of  AB  at  that  instant; 
or  of  xy  if  we  consider  x  and  y  as  the  variables.  Hence 
the  differential  of  the  product  of  two  variables  is  equal  to  the  sum 
of  the  products  arising  from  the  nmltiplication  of  each  va?'iable 
by  the  differential  of  the  other. 

Note. —  This  proposition  being  the  key  to  the  whole  subject  of  the  differential  cal- 
culus, should  be  carefully  studied  and  well  understood.  The  result  of  this  proposition 
might  have  been  surmised  by  considering  that  a  product  of  two  variables  is  subject  to 
two  independent  causes  which  produce  its  rate  of  change.  If  x  has  a  certain  rate  of 
increase,  that  of  the  product  will,  from  that  cause,  be  y  times  that  rate;  and  \i y  have  a 


DIFFERENTIATION    OF    FUNCTIONS. 


65 


D 

Y 

1 

^ 

certain  rate  of  increase,  that  of  the  product  will  from  that  cause  be  jr  times  that  rate  ; 
and  the  total  effect  of  both  causes  wi  1  be  the  sum  of  the  partial  effects  arising  from 
each  cause  independent  of  the  other;  that  is,  the  entire  rate  of  increase  of  xy^  is  y 
times  that  of  jjt  plus  x  times  that  oi  y.  This,  however,  is  not  mathematical proo/ — 
it  only  makes  the  result  probable. 

This  proposition  may  be  illustrated  geometrically  thus : 
Let  X  l)e  represented  by  the  line  AB  (Fig.  i),  and  y  by 
the  line  AC ;  then  the    product  xy  will  be  represented  by 
the  rectangle  ABDC.     Suppose     ,  F       o' 

X  and_y  to  be  each  increasing  in 

such  a  manner  that  when  x  has      '  ""  ~  I  ^ 

become  equal  to  AB^  and  is 
then  increasing  at  a  rate  that, 
if  continued,  would  produce  the 
increment  BB'  in  a  unit  of  time, 

y    will    have    become    equal    to     A  B        B' 

AC  and  be  increasing  at  a  rate  '^'  ^" 

that  would  produce  the  increment  CC  in  the  same  unit  of 
time.  Then  BB'  will  be  dx  and  CC  will  be  dy^  for  they 
will  l)e  the  true  symbols  of  the  rates  of  increase  of  .r  and  j. 
Now  the  uniform  increment,  BB\  of  x  will  produce  the 
uniform  increment  B  D  E  B'  of  the  rectangle,  which  will 
therefore  represent  its  rate  of  increase  arising  from  that 
of  X.  In  like  manner  C  C  F D  will  represent  the  rate  of 
increase  arising  from  that  of  y.  Hence  the  symbol  of  the 
entire  rate  of  increase  of  the  rectangle  arising  from  the  rates 
of  increase  of  the  sides,  will  be  the  sum  of  the  two  rectan- 
gles B  B'  E  D  and  C  C'  F D  ox ydx-\-xdy. 

It  must  be  remembered  that  the  increments  DE  and  DF 
are  ;?<?/ ^r/?/<:7/ increments  given  to  the  sides  BD  and  CD,  but 
suppositive  increments,  or  syiTibols^  representing  their  rates 
of  increase ;  for  in  order  that  the  suppositive  increment  of 
the  rectangle  may  be  uniform,  the  sides  BD  and  CD  must 
remain  constant.  Hence  the  two  rectangles  BB'ED  and 
CC'FD   are   not  actual  increments  of  the  rectangle  ABDC, 


66  DIFFERENTIAL    CALCULUS. 

but  suppositive  increments,  which  are  symbols  showing  what 
would  be  its  increment  if  the  rate  were  to  remain  constant 
from  that  point  in  the  value  of  x  and  y  for  a  unit  of  time,  and 
are,  therefore,  the  measure  of  that  rate. 

Note. — The  increment  of  the  rectangle  arising  from  the  actual  increments  of  x 
and  y  would  include  the  small  rectangle  DED'F  or  dx  .  dy  ;  and  hence  it  would  be  too 
great  to  represent  the  req\iired  rate  by  so  much  ;  and  it  is  to  get  rid  of  this  surplus  that 
the  advocates  of  the  infinitessimal  theory,  who  take  the  actu  <1  increment  to  represent 
the  r..te,  re  luce  it  to  an  infinitessimal,  which  they  claim  may  be  neglected  with  impu- 
nity. r>ut  we  see  the  true  reason  for  throwing  it  out  of  the  expression  for  the  rate  is, 
«£i/becau  e  of  its  insignificance,  but  because,  being  produced  by  the  actual  increments 
oix  and  y  after  they  have  passed  beyond  the  values  of  AB  and  AC,  it  has  no  connec- 
tion with  the  rate  at  which  the  rectangle  was  increasing  at  the  moment  x  and_y  were 
equal  to  those  values. 

Proposition  V. 

(13)  To  find  the  differential  of  the  product  of  any  number 
of  variable  factors. 

Let  us  first  take  the  product  of  three  variables  as 

xyz 
If  we  represent  the  product  xy  by  tt  we  shall  have 

X  y  z^=-uz 
and  (Art.  7  and  12) 

d{xyz)  =  ^(  uz)  =  zdu  -|-  iidz 
Replacing  u  by  its  value  we  have 

d{xyz)  ^^zd{xy)  -\-xydz 
or  since 

d{xy)  '=ydx  -f  xdy 
we  shall  have 

d{xyz)  =  zydx  -j-  zxdy + xydz 
If  we  take  the  product  of  four  variables  as 

xyzu 
and  make  xy=-r  and  zu-=^s  we  have 

xyzu^=^rs 
and 

di^xyzti)  ^=d(rs)  =rds~\-sdr 


DIFFERIlN'IIATIOX    OP'    FUNCTIONS.  67 

Replacing  ;-  and  s  by  their  values  we  have 
d{xy22/)  =^xyd{zti)  -\-zud{xy) 
substituting  the  values  of  d^zu)  and  d{x}>)  we  have 
di^xy  s  // )  =  xyzdu + xyudz  +  uzxdy + uzydx 

These  examples  may  be  carried  to  any  extent,  but  are 
enough  to  show  the  la-iu  which  governs  the  differential  of  a 
product,  which  law  may  be  thus  stated  :  The  rate  of  change 
in  a  product  arising  from  the  rate  of  any  one  factor  is  equal 
to  the  product  of  all  the  other  factors  multiplied  by  that 
rate ;  and  the  total  rate  of  change  in  the  product  is  the  sum 
of  all  the  partial  rates  arising  from  the  rates  of  the  factors 
taken  separately.  In  other  words,  the  total  effect  arising 
from  all  the  causes  acting  together  is  equal  to  the  sum  of 
the  partial  effects  arising  from  each  cause  acting  separately. 

Hence,  the  differential  of  the  product  of  any  number  of  vari- 
able factors  is  equal  to  the  sum  of  the  products  arising  from  7nul- 
tiplying  the  differential  of  each  variable  by  the  product  of  all  the 
other  variables. 

EXAMPLES. 


What  is  the  differential  of  ayz  ?  — 

A  ns .     aydz + azdy 
What  is  the  differential  of  4  bxy  ?  — 

Ans.      4  b{xdy -\-ydx) 
What  is  the  differential  of  ax-\-byz?  — 

A  ns .     adx  -\-  b\  'dz + bzdy 
What  is  the  differential  of  xy — //c? —  Ans. 
What  is  the  differential  of  '^ax — zxy  .■*  —  Ans. 
What  is  the  differential  of  2ay-{-;^i^?  —  Ans. 
What  is  the  differential  of  4abxyz  ?  —  Ans. 
What  is  the  differential  of  bcu  —  a'-^—zy? — Ans. 
What  is  the  differential  of  4axy  —  b'-^-\-cu? — Ans. 
What  is  the  differential  of  -a-  —  2bu-\-cy  ?  —  Ans^ 


A.v. 

I, 

Ex. 

2, 

Ex. 

3 

Ex. 

4 

Ex. 

5 

Ex. 

6, 

Ex. 

7' 

Ex. 

8, 

Ex. 

9 

Ex. 

\o 

68  DIFFERENTIAL    CALCULUS. 

Proposition  VI. 
(14)      To  find  the  differential  of  a  fraction. 

CASE    I. 

Let  the  fraction  be 

X 

n 
in  which   the  variable   is  divided   by  a  constant.      Make 

X 


-n=y 

then 

and  (Art.  lo) 

hence 

x=z  ny 
dx  =  ndy 

Jx\       dx 

cR, 


that  is, 

The  differential  of  a  fraction  having  a  vaj'iable  numerator 
and  a  constant  denominator  is  equal  to  the  differential  of  the 
numerator  divided  by  the  deno?ninator. 


CASE    2. 


Let  the  fraction  be 

fi 

X 

in  which 

the   denominator  is 

variable 

and 

the 

numerator 

constant. 

Ma»ke 

n  _ 

X 

~-y 

then 

and  (Art. 

12  and  Art. 

xy,= 
9) 

-  n 

hence 

d{xy) 

=  ydx  -f 
dv=  - 

xdy  = 
ydx 

;  dn 

=  0 

X 


DIFFERENTIATION    OF    FUNCTIONS.  69 

Replacing  _y  by  its  value  we  have 

^X^  X 


-dx  , 

ndx 


x' 


that  is, 

The  differential  of  a  constant  divided  by  a  variable  is  equal  to 
minus  the  numerator  into  the  differe?itial  of  the  denominator^ 
divided  by  the  square  of  the  denominator. 

CASE   3. 

Let  the  fraction  be 

X 

in  which  both  terms  are  variable,     Make 

X 

y 

then 

x^=^tiy 

and 

dx  ^=d{  uy)  ^ydu + udy 

hence 

-       dx — udv 
du=: — 

y 

Replacing  ti  by  its  value  we  have 

dx—dy 
,  ^'\ y       ydx — xdy 


'(^) 


Kyi  y  y 

that  is, 

The  differential  of  a  fraction  of  which  both  terms  are  vari- 
able^ is  equal  to  the  differential  of  the  numerator  imdtiplicd  by 
the  de?tomifiator,  minus  the  diffcre?itial  of  the  denominator  ?nul- 
tiplied  by  the  numerator^  and  this  difference  divided  by  the  square 
of  the  deno7?iinator. 

If  the  variables  in  any  of  these  cases  should  be  functions 
of  other  variables,  we  can  represent  them  by  single  letters 
and  replace  them  in  the  formula  by  their  values 

Thus,  suppose  the  given  fraction  to  be 


70 

make 
then 


DIFFERENTIAL    CALCULU: 

U  —  X 

Vy 

u  —  X  =^  s  and  vy  =i  r 


Replacing  the  values  of  s  and  r  we  have 

(u—x\ vyd{^  u—x)  —  (u— x)d{vy) 

d(  )— Ta 

\    7y  /  vy 

which  being  expanded  becomes 

'?/! — ^\ z^ydu — lydx — iwdy — tiv^v  -]-7>xdy  -j-xydv 


/u — ^\ 

4 — J 


vy  '  v^y 


EXAMPLES. 

Ex.    I.     What  is  the  differential  of  ax-j-j? 

0 


Ans.     adx  +-7- 
0 

Ex.    2.     What  is  the  differential  of  cxy-\ ? 

u 

adu 
Ans.     cxdy-\-cydx 2" 

Ex.  3.  What  is  the  differential  of  ^— ^?     ^f^s. 

Ex.  4.  What  is  the  differential  of  ^  ?     Ans. 

Ex.  ^.  What  is  the  differential  of  s^+^^?     Ans. 

Ex.  6.  What  is  the  differential  of  3(t?— ^)— |E^?     Ans. 

Ex.  7.  What  is  the  differential  of  2ax-\-x(y—c)?     Ans. 

Ex.  8.  What  is  the  differential  of  4((^_v^—x)(2^—<r)?     Ans. 

Ex.    9.  What  is  the  differential  of  ?=?X^^'?     Ans. 

Ex.  10.  What  is  the  differential  of  {a  —  i'){xy^z)  ?     A71S. 

Ex.  II.  What  is  the  differential  of  6ab—2,xy{7i—c)'?     Ans. 

Ex.  12.  What  is  the  differential  of  (2— _>')(<7— jc)-f-^?  Ans. 


DIFFERENTIATION    OF    FUNCTIONS.  71 

Proposition  V.I. 

(15)  To  find  the  differential  of  any  poiuer  of  a  va7'iable. 
We  have  seen  (Art.  13)  that  the  differential  of  the  product 

of  any  number  of  variables  is  equal  to  the  sum  of  the  pro- 
ducts arising  from  multiplying  the  differential  of  each  vari- 
able by  the  product  of  the  rest.  If  there  are  n  variable  fac- 
tors in  the  given  product,  there  will  be  71  products  in  the 
differential,  and  the  coefficient  of  the  differential  of  each 
variable  will  be  the  product  of  (/^— i)  variables.  If  now  all 
the  variables  become  equal,  we  may  represent  each  one  by 
X,  and  the  product  will  become  x^  ^  while  each  of  the  pro- 
ducts composing  the  differential  will  become  x^^^^dx;  and 
as  there  are  n  of  these  products,  the  sum  of  them  will  be 
nx^~^dx^  hence  we  have 

d  {x'')^nx"-^dx 

Hence,  the  differential  of  a  variable  raised  to  a  poiuer  is  equal 
to  the  variable  raised  to  the  s.nne  power  less  one^  and  midtiplied 
bv  the  exponent  of  the  poiver  and  the  differential  of  the  variable. 

If,  for  example,  we  take 

d  [xyzii)  ^=^ xyzdu  -\- xyudz-\- xzndy  -\-yzudx 
and  suppose  all  the  variables  to  become  equal,  we  shall  have 

dx^  ^=^  dfX^  dx 
If  we  represent  ;»:'*  by  y  we  shall  have 

dy=fLx^^'^dx 
and 

which  is  the  differential  coefficient  of  that  function  of  x. 

(16)  The  rule  here  given,  for  finding  the  differential  of 
the  power  of  a  variable,  holds  good  for  all  values  of  ;/, 
whether  integral  or  fractional,  positive  or  negative. 

Case  I.     Let  //  be  negative  and  represented  by  — ///,  then 

I 

H „— »l  — - 


72  DIFFERENTIAL    CALCULUS. 

and  (Art.  14) 

Dividing  both  terms  of  this  fraction  by  x'^^~^  we  have 

in  dx  _ 

-mx  ^  ^dx 


^?Ui-l 


in  accordance  with  the  rule. 

Case  2.     Let  n  be  equal  to  A  then 

^  m 

1_ 

Representing  :r"''  by  j^  we  have 

x=f'^ 
and 

or 


«-N 


Replacing  _y  by  its  value  we  have 


dW^  /    IX 


■  —  — jr 


Ci^j'<f  3.      Let  ;z  be  equal  to  "^^  and  represent  .r"''  by  y  and 

we  have 

_r 

j=.t"''  or  _>'™=Jt:'* 

and 

whence 

j'x^—'^dx 

Replacing  J/  by  its  value,  we  have 
hence  the  rule  is  true  in  all  cases. 


DIFFERENTIATION    OF    FUNCTIONS.  75 

Proposition  VIII. 

(17)  To  find  the  diffei-ential  of  any  root  of  a  variable. 

Let  us  take  the  function'^/^  and  make  it  equal  to  yy 
then 

y'^^^zzx 

and 

my^~^dy^=^dx 
whence 

dx 

Replacing/  by  its  value  we  have 

dx 

That  is,  the  differential  of  any  root  of  a  variable  is  equal  ta 
the  differential  of  the  variable  divided  by  the  index  of  the  roof 
into  the  same  root  of  the  variable  raised  to  a  pouier  denoted  by  the 
index  of  the  root  less  one. 

Hence 

_        dx 

d\/  x=- 7=- 

These  results  could  of  course  be  obtained  under  the  pre- 
ceding rule,  by  giving  to  the  variable  a  fractional  exponent. 
But  this  is  not  always  convenient. 

(18)  In  all  the  cases  which  have  been  explained  under 
these  eight  propositions,  the  single  letters  that  have  been 
used  may  represent  functions  of  other  variables,  in  which 
case  the  operation  must  be  continued  by  substituting  the 
functions  for  the  letters  representing  them  and  performing 
upon  the7n  the  operations  indicated,  which  can  be  done  by 
the  rules  already  given ;  for  the  terms  of  all  algebraic  func- 
tions must  ultimately  take  some  one  of  the  forms  that  we 
have  discussed. 


74  DIFFERENTIAL    CALCULUS. 

Thus,  if  we  have  the  function 

a  —  x 
we  may  represent  the  numerator  by  ti  and  the  denominator 
by  i>  we  shall  then  have 


•^^  —  V  y     ti 


a  —  X         V 
and 

^x^  —  y^'S      vdu—udv 


^^-. 


>•    a  —  X    / 


^2 


Replacing  tc  and  v  by  their  values  we  have 

^(^^~y  y\  _ (<^?  —  x)d{x^  —  yj)  —  {^^  —  y ~y)d{a  —  x) 
V    <;?  — ji-    /  i^a — xY' 

and  performing  the  operations  indicated  we  have 

^  x--y~v\_{^-^){^^^^'^'^-^^)+{^'^-y~})^^^ 

\     a  —  x     /  {a—xY 

which  completes  the  differentiation. 

Proposition  IX. 

(19)  To  find  the  differential  of  a  function  with  respect  to 
the  independent  variable  when  it  enters  the  given  function 
by  means  of  another  function  of  itself. 

Let  there  be  a  function  of  j'  in  which  y  represents  a  func- 
tion of  X.  The  proposition  is  to  find  the  differential  co-effi- 
cient of  the  given  function  with  respect  to  x.  Represent- 
ing the  given  function  by  u  we  have 

u^=F(y)  SiXid y^F{x) 

In  order  to  find  the  relation  between  du  and  dx,  it  might 
be  supposed  necessary  to  eliminate  j  from  the  equations  and 
obtain  one  directly  between  zi  and  .t,  to  which  the  ordinary 
process  of  differentiation  could  be  applied.     But  this  is  not 


DIFFERENTIATION    OF    FUNCTIONS.  75 

necessary ;  for  if  we  differentiate   these  equations   as   they 
are  we  have 

dit=F'{y)dy  and  (iy=F\x)(i^x 
in  which  F'  {y)  and  F'  {x)  represent  the  differential  coeffi- 
cient of  zi  with  respect  to  y  and   of  y  with  respect  to  x. 
Fro:n  the  first  of  these  we  obtain 

dy  =  J^ 
and  placing  this  value  of  dy  equal  to  the  other  we  have 
-Y^{-F'{x)dx  or  S=^'(7)  •  ^\x) 

That  is,  the  differential  coefficient  of  2c  with  respect  to  x 
is  equal  to  that  of  //  with  respect  to  _y  multiplied  by  that  of  j/ 
with  respect  to  x.     Hence 

When  the  variable  of  a  given  function  represents  the 
function  of  an  independent  variable,  then  the  diff'erential 
coffficie7it  with  respect  to  the  independetit  variable  is  equal  to  the 
product  of  the  differential  coefficient  of  the  function  with  respect 
to  the  given  variable^  multiplied  by  the  differential  coefficient  of 
the  given  variable  with  respect  to  the  independent  variable. 

Thus  if  we  have  the  function  ^^ — ay  in  which  _)'=2^z — x^ ^ 
representing  the  given  function  by  ?/,  we  have 


whence 


-_2y-a  and  a-^--2x 


du 


■^'=^2ax — 4xy^4x^  —  Gax 


EXAMPLES. 

Find  the  differentials  of  the  following  functions  in  which 
the  variables  are  independent. 

Fx.     I.     a^x^-{-z  Afis.      2a^xdx-\-dz 

Fx.     2.     bx^  —y'^-\-a  Ans.     2bxdx—2,y"dv 

Fx.     3.     ax^  —bx^-{-x  Ans. 

Fx.     4      (^+'/)(i'~-^')  ^ns. 


7^  DIFFERENTIAL    CALCULUS. 

Ex.     5.  ^x^-2ay-b^  Ans, 

Ex.     6.  x'^-x'^j^^Ij  ^^^^_ 

Ex.    7.  ax'^--^bx  Ans. 

Ex.     8.  [x'^^a){x-d)  4ns. 

Ex.     9.  x^y^—z'^  Ans. 

Ex.  10.  ax^{x^  -\-a)  Ans. 

^^-    II-         ^_2j;2  Ans. 

Ex.    12.       V,^2_^2  ^^^^ 

Ex.  16.     (a  +  V^)^  ^;z^. 

^'^-  i^-       T-L-v-n  Ans. 

Am. 

Ans, 

Ans. 

Ans, 

Ex.  2T,.     x~^y-^  4,js 

Ex.24.     7nx~^+{x^yY  Ans. 

Ex.  25.     x-y^-z  Ans. 

Ex.  2Xi.     a-^bcx—T^v"^  \,js^ 

Ex.2'].     x-^^y^J^z^  i^^s. 

Ex.  28       {ax-yY  :,^s. 


DIFFERENTIATION    OF    FUNCTIONS.  77 

Ex.  29.   {b  —  c){x—}^-  Alls. 

Ex.  30.   Vjc^-H^a/^  ^'^•^• 

75".^.  31.  A  person  is  walking  towards  the  foot  of  a  tower, 
on  a  horizontal  plain,  at  the  rate  of  5  miles  an  hour;  at  what 
rate  is  he  approaching  the  top,  which  is  60  feet  high,  when 
he  is  80  feet  from  the  bottom  ? 

Let  the  height  of  the  tower  be  a.,  the  distance  of  the  per- 
son from  the  foot  of  it  be  x.^  and  the  distance  from  the  top 
be  J';  then 

and 

ydy^^xdx 

whence 

Ans.     4  uiilcs  per  hour. 

Ex.  32.  Two  ships  start  from  the  same  point  and  sail, 
one  north  at  the  rate  of  6  miles  per  hour,  and  the  other  east 
at  the  rate  of  8  miles  per  hour;  at  what  rate  are  the  ships 
leaving  each  other  at  the  end  of  two  hours  ? 

A?is.      10  1/iiles  per  hour. 

Ex.  2iZ-  Two  vessels  sail  directly  south  from  two  points 
on  the  equator  40  miles  apart  ;  one  sails  at  the  rate  of  5 
miles  per  hour,  and  the  oilier  at  the  rate  of  10  miles  \)tx 
hour;  how  far  will  they  be  apart  at  the  end  of  6  hours,  and 
at  what  rate  will  they  be  separating  from  each  other,  suppos- 
ing the  meridians  to  be  parallel?         A.ns.     3  miles  per  hour. 

Ex.  34;  A  ship  sails  directly  south  at  the  rate  of  10 
miles  per  hour,  and  another  ship  sailing  due  west  crosses  her 
track  two  hours  after  she  has  passed  the  point  of  crossing,  at 
the  rate  of  8  miles  per  hour;  at  what  rate  are  they  leaving 
each  other  one  hour  afterwards? 

Ans.     11.74  miles  per  hour  nearly. 

Ex.  35.     The  vessels  sailing  as  in  the  last  case,  how  wib 


78  .  DIFFERENTIAL    CALCULUS. 

the  distance  between  them  be  changing,  and  at  what  rate, 
one  hour  before  the  second  crosses  the  track  of  the  first  ? 

At  that  time  the  first  vessel  will  be  lo  miles  from  the  point 
of  crossing,  and  the  second  8  miles.  Calling  the  distance 
of  the  first  x,  and*  the  second  j^,  the  distance  between  them 
is  \^ x^-\-y^  which  we  will  call  ?/,  then 

and 

xdx  -\-  ydy      loo  —  64 


du=^- 


\/x^-^y^    v"  100+64 

We  make  64  in  the  numerator  negative  because  j  is  a  posi- 
tive decreasing  function,  and  its  differential  is  therefore 
intrinsically  negative. 

From  the  above  equation  we  find  that  the  vessels  are  sep- 
arating at  the  rate  of  2.812  miles  per  hour  nearly. 

£x.  ^6.  The  height  of  an  equilateral  triangle  is  24  inches, 
and  is  increasing  at  the  rate  of  two  inches  per  day ;  how 
fast  is  the  area  of   the  triangle  increasing? 

Ans.     32^3  square  inches. 

Ex.  37.  The  diameter  of  a  cylinder  is  2  feet,  and  is  in- 
creasing at  the  rate  of  i  inch  per  day,  while  the  height  is 
4  feet  and  decreasing  at  the  rate  of  2  inches  per  day ;  how 
is  the  volume  changing.''  and  how  the  convex  surface.'' 

Ans.  The  volume  is  increasing  at  the  rate  of  288  r.  cubic 
inches  per  day.  The  area  of  the  convex  surface  is  not 
changing. 


SECTION     III. 


SUCCESSIVE   DIEFEREMTIALS. 

(20)  In  considering  the  differential  of  a  function  hitherto, 
we  have  regarded  it  as  immaterial  whether  that  of  the  inde- 
pendent variable  was  itself  variable  or  uniform.  In  consid- 
ering the  rate  of  change  in  the  differential  of  the  function, 
it  will  be  most  convenient  to  consider  that  of  the  indepen- 
dent variable  as  constant ;  and  this  we  have  a  right  to  do, 
since,  the  variable  being  independent,  its  rate  is  always 
assignable. 

If  we  have  the  product  of  two  or  more  independent  vari- 
ables, whether  they  are  alike  as  x'^ ,  or  different  as  x.y.z^  the 
value  of  the  rate  of  change  depends  not  only  on  that  of  each 
independent  variable,  but  also  on  the  absolute  value  of  all 
the  variables.  Thus  the  value  of  ^/(jc^),  or  3jc-//^,  depends 
not  only  on  that  of  dx^  but  also  on  the  absolute  value  of  jc^, 
and  is  greater  or  less  as  x"^  is  greater  or  less.  So  that  T^x'^d^ , 
which  is  the.  rate  of  change  of  .r^,  has  its  own  rate  of  change 
or  differential. 

To  find  this  second  differential  we  must  treat  the  first  as 
an  original  function;  and  as  .v  is  supposed  to  change  uni- 
formly, dx  will  be  regarded  as  a  constant.  Now  the  func- 
tion T^x'^dx  may  take  the  form  3^jc.:r-,  and 

^3<^/-T.jc^):=  3^/.v.c/(.r")=:  3c/.T,2.r^/a;=:  dxdx" 
This  is  called  the  second  differential  of  jv:^,  being  the  differ- 

79 


So  DIFFERExNTIAL    CALCULUS. 

ential  of  the  differential.  This  order  is  indicated  by  plac- 
ing the  figure  2  as  a  sort  of  exponent  to  the  letter  d,  thus 
.^^(x^)  is  the  symbol  for  the  second  differential  of  :v^,  and 
hence 

If  the  function  is  at  all  complicated,  and,  especially,  if 
we  desire  to  indicate  a  differential  coefficient,  it  is  much  mor  j 
convenient  to  represent  it  by  a  single  letter;  in  which  case 
the  letter  itself  is,  for  the  sake  of  brevity,  called  the  func- 
tion. 

So  that  if  we  represent  x"^  by  u  we  shall  have 

?/  =  .t3  and  d"u=:6xdx^  (i) 

Since  the  second  member  of  this  last  equation  still  con- 
tains the  letter  x,  it  will  still  have  a  rate  of  change  which 
may  be  found  by  considering  6dx'^  as  the  constant  coefficient 
of  X,  and  difTerentiating  we  have  (Art.  10) 

d^u^=^6dx^  .dx=:6dx^  (2) 

The  expressions  dx"  and  dx^  are  to  be  understood  as 
indicating,  noi  the  differentials  of  x"  and  x^,  but  the  square 
and  cube  of  dx. 

The  figure  placed  as  an  exponent  to  the  letter  d  indicates 
the  o?'de?'  of  the  differential;  and  the  differential  of  any  order 
above  the  first  is  the  rate  of  change  in  the  differential  of  the 
previous  order.  The  differentiation  of  the  differential  can 
take  place  only  while  the  latter  represents  what  is  still  a 
function  of  the  independent  variable.  The  differential  of 
an  independent  variable,  being,  as  we  have  stated,  supposed 
to  be  constant,  can  have  no  differential. 

If  we  divide  equations  (i)  and  (2)  respectively  by  dx^ 

and  dx^,  we  have 

d^t/       ^            J     d^u       r 
- — ~^=.bx   and  =  6 

dx'^  dx^ 

in  which  the  second  members  are  the  second  and  third  dif 
ferential  coefficients  of  the  function  x^ . 


SUCCESSIVE    DIFFERENTIALS 


(21)  To  illustrate  the  principle  of  successive  differentia- 
tion, let  us  suppose  A  B  D  C F  E  G  (Fig.  2)  to  represent  a 
cube,  of  which  the  side  A  B  is  an  increasing  variable  repre- 
sented by  X.  Let  the  suppositive  increments  Dd^  Dd'  and 
Dd"  represent  the  three  equal  rates  of  increase  of  the  three 
x's  whose  product  is  the  given  cube ;  we  are  to  find  what 
will  be  the  corresponding  rate  of  increase  of  the  cube  itself. 
That  is,  Z>^/ being  equal  to  dx,  what  is  the  value  of  d{x^^. 

At  the  moment  the  sides  of  the  cube  become  equal  to  x^ 
the  cube  tends  to  expand  by  the  movement  of  the  three  faces 
DA^  B>F  ^nd  DG  outward  in  the  directions  Dd,  Dd'  and 
Dd",  each  face  continuing  parallel  to  itself,  and  thus  in- 
creasing the  cube.  But  in  order  that  these  increments  may 
represent  the  rate  at  which  the  cube  was  increasing  when  they 
began,  they  must  be  mad'e  iinifonnly  at  the  same  rate. 
Hence  the  areas  of  the  faces  must  remain  constant,  and  they 
must  move  at  the 
same  rate  as  the  in- 
crements Dd,  Dd ' 
and  Dd"  are  des- 
cribed ;  the  move- 
ment being  con- 
trolled by  those  in- 
crements. Thus  the 
three  solids  Da^  Df 
and  Dg,  generated 
by  the  flowing  out 
of  the  surfaces  DA, 
DF  and  DG,  with 
an  unchanging  area, 
at  the  same  rate  as 
wlien  their  sides  be- 
came    equal     to    x, 

form    the  increment 
6 


Fig.  2. 


82  DIFFERENTIAL    CALCULUS. 

that  would  take  place  in  the  cube  in  a  unit  of  time,  at  the  rate 
at  which  it  was  increasing  when  its  side,  or  edge,  became 
equal  to  x\  and  hence  they  form  the  true  symbolic  incre- 
ment which  represents  the  rate  of  increase  or  differential  of 
x"^ .       Now  each  of  the  solids  thus  formed  is  equal  to  x^dx^ 

and  hence 

d{x^^^=-';iyXrdx 

But  if  the  cube,  when  its  edge  is  equal  to  x^  is  in  a  state 
of  increase,  the  faces  DA^  DF  and  DG  have  other  tenden- 
cies besides  that  of  flowing  directly  outward.  They  also  tend 
to  expand  in  the  direction  of  their  own  sides,  and  this  ten- 
dency is  quite  distinct  from  the  other.  Let  us  examine  and 
measure  it.  The  tendency  of  the  face  DF  to  expand,  aris- 
ing from  that  of  the  cube  itself,  would  be  by  the  flowing  out 
of  the  sides  DC  and  DE  in  the  directions  Dd  and  Dd\  and 
remaining  parallel  to  themselves.  The  rate  of  increase  of 
the  face  Z>7^  would  be  measured  by  the  areas  described  by 
these  flowing  lines  in  a  unit  of  time,  at  the  same  rate  as 
when  they  became  equal  to  x^  their  lengths  being  constant; 
that  is,  by  the  areas  Dc  and  De  -  But  the  face  DF  of  tne 
cube  is  the  base  of  the  solid  Df,  and  the  tendency  of  the 
base  to  expand  imparts  a  like  tendency  to  the  solid,  and  the 
rate  at  which  the  solid  tends  to  expand  is  such  that  while 
the  base  would  increase  by  the  rectangles  Dc  and  De',  the 
solid  would  increase  by  the  solids  Dc  and  De\  which  there- 
fore represent,  symbolically,  the  rate  of  increase  of  Df. 
Now  Dc'  and  De"  are  each  equal  to  xdx'^,  and  hence  the  rate 
of  increase  of  Df  is  equal  to  2xdx^.  But  the  differential 
of  the  cube  is  represented  by  three  such  solids  as  Df  and 
the  rate  of  the  increase  of  the  whole,  or  the  differential  of  the 
differential  of  x"^  is  dxdx'^. 

Again  the  solid  D(f  or  xdx^  tends  to  increase  in  the  direc- 
tion Dd'  at  such  a  rate  as  would  generate  the  suppositive 
increment  Dd"\  equal  to  dx^  in  the  same  unit  of  time  and 


SUCCESSIVE    DIFFERENTIALS.  85 

in  a  uniform  manner.     Hence  d{x(ix^)  is  equal  to  dx"^^  and, 
therefore,  d{6xdx'^)  is  equal  to  Gdx^. 

(22)  It  must  always  be  remembered  that  the  solids  rep- 
resented in  the  figure  are  Jiol  the  actual  increments  of  the 
cube,  but  the  symbols  which  represent  its  ?'afc  of  increase  and 
the  successive  rates  of  that  rate  at  the  instant  that  x  is  equal 
to  AB;  that  is,  they  are  the  increments  that  would  take  place 
in  the  cube,  and  in  the  increments  themselves  if  made  uni- 
formly. The  hViU  which  governs  the  increase  of  the  cube, 
contains  within  itself  not  only  the  rate  at  which  the  cube  is,, 
at  any  instant  increasing,  but  also  the  law  of  change  in  that 
rate,  and  the  law  to  which  that  law  is  subject ;  and  these 
symbols  represent  the  development  of  that  law  which  was 
actually  operating  at  the  instant  the  cube  attained  the  value 
of  AB'^  or  x^,  and  before  any  farther  increase  had  taken 
place. 

(23)  We  may  learn  from  this  demonstration  the  method 
by  which  the  actual  increment  of  a  power  may  be  developed. 
By  dissecting  the  figure  (Fig.  2)  and  noticing  the  parts  of 
which  the  increased  cube  is  composed,  we  find,  firsts  the 
original  cube  or  x^ ;  second^  the  three  solids  Da^  Df  and  Dg 
or    T^x^dx;    thirds  the  three  solids  Dc\  De"   and  Db  which 

6  'Xd'K 

represent  half  \\\q  rate  of  increase  of  T^x'^dx  or  — — ^— ;  and 

2 

fourth^  the  solid  or  small  cube  Da"\  which  represents  one- 

6  xdx  ^dx 

third  of  the   rate  of  increase  of  — — "—  or   — 1— ;  and  these 

2  2.3 

make  up  the  volume  of  the  cube  after  being   increased  hy 

the  addition  arising  from  the  increment  dx  to  the  side  AB. 

But  these  increments  are  suppositive,  and  are  used  merely 

as  symbols  to  show  the  successive   rates  of  increase,  all  of 

which  exist  in  the  function  .v"^  before  any  increment  actually 

takes  place. 

If  we  divide  -^x^dx  by  dx  we  shall  have  the  first  differen- 


84  DIFFERENTIAL    CALCULUS. 

tial  coefficient  of  x^.     If  we  divide by  dx^  we  shall 

2 

have  half  the  second  differential  coefficient  of  x^ ;  and  if 

•   •         ^cix  o  •  • 

we  divide   by  dx^  we  shall  have  one -sixth  of  the  third 

2  ■  3 
differential  coefficient  of  x^ \    and   these   results,  viz.:   -^x"^^ 

— -and are  the    coefficients    by  which  the    successive 

2  2.3 

powers  of  dx  must  be  multiplied  in  order  to  make  up  the 
parts  composing  the  suppositive  increment  of  x^.  Now 
since  these  partial  increments  taken  together  with  the  orig- 
inal cube  form  also  a  complete  cube,  if  we  make  dx  a  real 
increment  and  multiply  its  successive  powers  by  these  same 
coefficients,  we  shall  have  an  actual  increment  of  the  cube, 
and  the  original  x^  will   have  become  {^x-\-dxY'. 

It  must  not  be  forgotten  that  these  differential  coefficients 
are  true  of  the  cube  before  the  increment  takes  place,  and 
when  dx  is  equal  to  zero 

For  convenience  we  will  designate  the  variable  cube  by 
u^  and  in  order  to  mark  the  point  where  the  differential  is  to 
be  taken  we  represent  its  variable  edge  by  {Ji-\-x^  in  which 
h  represents  the  side  AB,  or  that  particular  value  of  the 
variable  where  the  differential  is  to  be  taken,  while  x  will 
represent  its  variable  increment  and  take  the  place  of  dx. 
Then  the  cube  AE  or  ~aR'  will  be  represented  by  u  or 
{h-\-x^^  at  the  time  when  the  variable  u  equals  h  or  x^o. 

This  being  premised  we  take  the  differential  coefficients 
already  found,  namely,  3jc^,  f  and  g^,  and  substitute  {h-\-x^ 

for  X  (reducing  x  to  zero  at  the  same  time)  and  x  for  dx  in 
the   other  factors ;    then   3:^'^  becomes  the  first  differential 

coefficient  of  ti  or  (/^4-.^)^,  that  is  "d^=3(/^+^)^  with  x^=o^ 

or  3/r;  l"^  becomes  — 5—"  with  x^o;  that  is  3//,  or  half  the 

second   differential   coefficient  of  u  with    x-=io ;    and   373 


SUCCESSIVE    DIFFERENTIALS.  85 

becomes  one-sixth  of  the  third  differential  coefficient  of  u  or 


Hence,  indicating  by  a  vinculum  that  x  has  been  made 
equal  to  zero,  we  have  for  the  three  coefficients  3x"^,  -2  and 
,  which  were  true  of  the  cube  before  any  increment  was 


2  .  3 

made, 

also  true  of  the  cube  at  the  same  time  ;  and  the  different  parts 
of  the  cube  increased  will  be  represented  as  follows : 

The  cube  AE  by  {u)  or  /r^. 

The  three  solids  Da,  Df  Dg  by  {^^)x  or  2>Ji^x. 

The  three  solids  JDc ,  De '  and  Db  by  (     ,  ^j^T-  or  3//X-. 

The  solid  Dd'  "  by   (^ T^i)^    or  ^  -, 

and  these  make  up  the  value  of  (//+a:)'^  hence 

(dii\  (d'^  u\    ^^     f     d^u     \ 

uor  {^^+^Y  =  {^^)  +  \j:x)  ""  +\ld^^)''~'^\2.  2,.  dx^)"^ 
or 

This  illustrates  to  some  extent  the  law  which  connects 
together  the  parts  which  go  to  make  up  the  change  in  the 
function  of  a  variable  arising  from  that  of  the  variable  itself. 
A  more  complete  and  general  demonstration  of  this  law  is 
contained  in  the  following  theorem, 

Maclaurin's  Theorem. 

(24)  A  function  of  a  single  variable  may  often  be  ex- 
panded into  a  series  by  the  following  method. 


86  DIFFERENTIAL    CALCULUS. 

Representing  the  function  by  ti  and  the  variable  by  x  we 
shall  have 

When  this  function  can  be  developed,  the  only  quantities 
that  can  appear  in  the  development,  besides  the  powers  of 
J?,  will  be  constant  terms  and  constant  coefficients  of  those 
powers.  Hence  the  developed  function  may  be  put  into 
the  following  form  : 

u^A-\-Bx^Cx--^Dx^-\-Ex'^^  etc.  (i) 

in  which  A^  B^  C,  /?,  E,  etc.,  are  independent  of  x.  The 
problem  is  to  find  the  value  of  this  constant  term  A^  and 
the  values  of  the  constant  coefficients  B,  C,  Z>,  E,  etc.  For 
this  purpose  we  differentiate  equation  (i)  successively,  and 
divide  each  result  by  dx^  the  successive  differential  coeffi- 
cients will  then  be 

du  ^ 

-TZ=B-{-2Cx-\-T^Dx''-\-4Ex'^-\-  etc.  (2) 

-^2-  =  2C+2  .  3  .  E>x+s  •  4  .  ^^T~-f  etc.  (3) 

d^  u 

^  =  2  .  3  .  E>-}-2  .3.4.  Ex-i-  etc.  (4) 

Since  x  is  an  independent  variable,  these  equations  are 
true  for  all  values  of  x,  and,  of  course,  when  x^^o. 

Reducing  x  to  zero  in  equation  (i),  A  becomes  equal  to 
the  original  function  with  x  reduced  to  zero.  We  will  rep- 
resent that  state  of  the  function  by  {?i) ;  and  also  indicate  by 
brackets  around  the  differential  coefficients  that  A:=^in  f/iei'r 
values  also.  Then  from  equations  (2),  (3),  (4),  etc.,  we  have 
^     /^^M       ^     (d^ii\  (d'^ic\ 

and  so  on  to  the  end  of  the  development,  if  it  can  be  com- 
pleted ;  if  not,  then  in  an  unlimited  series. 

Substituting  these  values  of  A^  B,  C,  Z>,  etc.,  in  equation 
(i)  we  have 


SUCCESSIVE    DIFFERENTIALS.  87 

which  is  Maclaurin's  theorem. 

EXAMPLES. 

Ex.  I.     Expand  {a-\-xy^  into  a  series. 
Represent  {a-^xY'  by  u  and  we  shall  have 

u^i^a^xY^A^-Bx^Cx^^-Dx'^^-  etc.  (i) 

Differentiating  we  have 

du         ,         ,     , 

^=;.(;2-i)(.7+.Tr-2 
-^:=^n{n-i){n-2){a-\-xY-'^ 

--^=n{n-\){ii-2\n-i){a^-xY-'^ 

from  which,  when  ^=<?,  we  have 

A  =^  a"" 
B  =  nd"-'^ 

C  =  'A!!ll2)a^^ 


D 


_  ;;(//-!)(;/— 2)  ^^^3 


2  .3 

-ft  =  

2  ■  3  •  4 

Substituting  these  values  in  equation  (i)  we  have 

and  so  on ;  the  same  result  as  by  the  binomial  theorem. 
Ex.  2,     Develop  ^r;r^  into  a  series. 
We  have  by  differentiation 


DIFFERENTIAL    CALCULUS. 

du I 

dx  (^-f-jf)^ 

d^  u  2 


dx"      {^Li-\-xY' 
d^  u  2.3 


dx'^  {^a-[-xY 

and  so  on. 

Making  x=^o  in  the  values  of  //  and  the  differential  coef- 
ficients we  have 

i_     (^^^\_       I        /^^?/\_  2       (^^^^^\_     2.3 
^^''~a'    vT^y        "^'    Vd^)-~^'    \~d^)~~~^ 
Substituting  these  values  in  Maclaurin's  formula  we  have 
I         I        X    .   x"^       x^    , 

Ex.  3.     Develop  the  function  -ji:^  into  a  series. 

Ans.      i-\-x-\-x'^-\-x^-\-x'^-{-  etc. 
Ex.  4.     Develop  the  function  ^/  a+x  into  a  series. 

1  _i  1        __3_  1     o        _o_ 

Ans.     a^ -\-\a  ^^x—^-^a   ^x^-^^rir^d  ^.^ic'^— etc, 

I 
Ex.  5.     Develop    /     ,     x^  into  a  series.      Ans. 

Ex.  6.     Develop  ^'\/ {a+xf>  into  a  series.     Ans. 

Ex.  7.     Develop  ^a^^bx  into  a  series.     ^;^i'. 

I 

Ex.  8.     DeveloD     /  ,         into  a  series.     y^/2.f. 

^:r.  9.     Develop  («^—^^)   ^  into  a  series.     A7ts. 

The  formula  of  Maclaurin  applies  in  general  to  all  the 
functions  of  a  single  variable  that  are  capable  of  successive 
differentiations.  But  there  are  cases  in  which  the  function 
or  some  of  its  differential  coefficients  become  infinite  when 
x^=oj  in  such  cases  the  formula  will  not  apply.     The  func- 

tion,  c-{-ax'^  is  an  example  of  this  kind  ;  for  if  we  represent 
it  by  u  we  have 


SUCCESSIVE    DIFFERENTIALS.  89 


1 


and 

dt^      1      -• J  _  ^ 

If  in  this  we  make  x=o  for  the  value  of  the  coefficient 
B,  we  have 

In  general  no  algebraic  function  of  x  in  which  .v  is  not  con- 
nected with  a  constant  term  under  the  same  exponent,  can  be 
developed  by  this  theorem;  for  the  differential  coefficients 
will  be  such  as  to  reduce  to  zero  or  infinity  in  every  case, 
when  X  is  made  equal  to  zero. 

Taylor's  Theorem. 

(25)  T/ie  object  of  this  theorem  is  to  obtain  a  formula  for  the 
development  of  a  function  of  the  sum  or  difference  of  two  vari- 
ables. 

The  principle  on  which  this  theorem  is  based  is  the  fol- 
lowing :  The  differential  coefficient  of  a  function  of  the  sum 
or  difference  of  two  variables,  will  be  the  same  whether  the 
function  is  differentiated  with  respect  to  one  variable  alone, 
or  to  the  other  variable  alone.  Thus  the  differential  co- 
efficient of  {x-\-yy^  will  be  ;/(jc4-r)'^~^  if  we  differentiate 
with  respect  to  either  variable  alone,  the  other  being  con- 
sidered as  constant. 

A  function  of  the  sum  or  difference  of  two  variables  is  one 
in  which  both  are  subject  to  the  same  conditions,  so  that  the 
value  of  their  sum  or  difference  might  be  expressed  by  a 
single  variable  without  otherwise  changing  the  form  of  the 
function  ;  and  hence  we  may  regard  this  sum  or  difference 
as  itself  a  single  variable.  Now  any  rate  of  change  in  one 
of  the  two  component  parts  (the   other  being   regarded    as 


90  DIFFERENTIAL    CALCULUS. 

constant)  will  produce  the  same  rate  in  the  compound  vari- 
able (so  to  speak)  as  it  has  itself;  thus  :r+_;/ will  increase  at 
the  same  rate  as  x  if  y  be  constant,  and  at  the  same  rate  as 
j>  [{  X  be  constant.  So  that  changing  from  one  to  the  other 
is  merely  changing  the  rate  of  the  single  variable  that  would 
represent  the  value  of  their  sum  or  difference.  But  such 
change  in  the  rate  will  not  change  the  7'afw  which  it  bears 
to  the  corresponding  rate  of  the  function  (Art.  6)  ;  that  is,  it 
will  not  affect  the  differential  coefficient. 

(26)  To  apply  this  principle  let  us  take  any  function  of 
the  sum  of  two  variables,  as  J^{x-\-y),  which  we  will  repre- 
sent by  u.  If  it  can  be  developed  into  a  series,  the  terms 
of  the  series  may  always  be  arranged  according  to  the  powers 
of  y  ;  the  coefficients  being  functions  of  x  and  the  constants ; 
hence  it  may  be  made  to  take  the  following  form: 

u=j7(^^^y)=^^By-\-Cy^+By^+£y'^-\-  etc.         (i) 

in   which   A,  B^  C,  D,   £,  etc.,  are   independent  of  _y,  but 
functions  of  x. 

If  we  differentiate    equation   (i)  regarding  y  as  constant, 
and  divide  by  tfx,  we  shall  have 

du__dA     dB_       dC        cW  ^ 
dx~dx     dx-^      dx^       dx-^ 


If  we  regard  x  as  constant,  and  divide  by  dy,  we  have 
dii 
dv 


■=B+2Cy-^Sl?y^+4£y^+  etc. 


and  since  ^^  is  equal  to  jy  the  second  members  of  these 
equations  are  equal ;  and  since  this  equality  exists  for  every 
value  of  J,  and  since  the  coefficients  are  independent  of  that 
value,  the  corresponding  terms  containing  the  same  powers 
of  y  must  be  equal  each  to  each ;  hence 


SUCCESSIVE    DIFFERENTIALS. 


91 


^=^  (^) 

dB 

^^'^  (3) 

dC 

^-3^  (4) 

dB 

If  now  we  make  j'=<?,  then  F{x  +  y)  becomes  F(x),  which 
Ave  will  represent  by  z.  Under  this  supposition  equation  (i) 
will  become 

u  (now  become  z)  =^A 

Substituting  this  value  of  A  in  equation  (2)  we  have 

a^  —  -^ 
Substituting  this  value  of  JJ  in  equation  (3)  we  have 

dx     dx^  ~ 

whence 

d'^z 

2(LX'' 

similarly  we  have 

d'^z  _  d^z 

2dx^     ^  2  .  T,  .  dx^ 

and 


o 


j-4=a/r  or   E=^ 3-7 

2.3.  dx^  2.3.4.  dx^ 

and  so  on. 

Substituting  these  values  in  equation  (i)  we  have 

dz        d-z     ^  d'^z 

u  =  F{x+y)=z+^y-\-jj^,f+j-j-^,y^Jr  etc. 

in  which  the  first  term  is  what  the  function  becomes  when 
;•=''',  and  all  the  coefficients  of  the  powers  of  _y  are  derived 
from  it  on  the  same  supposition. 


92  DIFFERENTIAL    CALCULUS. 

This  is  the  formula  of  Taylor. 

A  function  of  x—y  is  developed  by  the  same  formula  by 
changing  y  into  —y,  thus  : 


EXAMPLES. 

£x.  I     Let  it  be  required  to  develop  (^+J^)"'' 
Representing  this  function  by  u  we  have 
u^=^(^x-\-yy^  and  z^^x^ 
then  by  differentiation 

^jc  '  dx'       ^  '  dx'       ^         ^^         ' 

Substituting  these  values  in  the  formula  we  have 

etc.,  the  same  as  by  the  binomial  theorem. 
Ex.  2.     Develop  the  function  V-^+j- 


L  i  — 1  _3.    ^  _5_ 

^;^i-.    (jc+j')^=''^^+i^^  ^-^~2^'^  ^-^""^2 ^4^6  ^  ^7^— etc. 
^jc.  3.     Develop  ^x+y. 

Ans.     .r-^+Jjc  ^7— -^^-^  ^^^ "'■^i^^'^'   ^j'^— etc. 

Ex.  4.      Develop  the  function  (x—y)'^.     Ans. 

3. 
^jc.  5.     Develop  the  function  {x—y)".     Ans. 

Ex.  6.     Develop  the  function  ^zr:^.     Afzs. 

_2 

Ex.  7.     Develop  the  function  of  a{x—y)   ^.     Ans. 

(27)  Although  a  function  of  the  sum  or  difference  of  two 
variables  can  generally  be  developed  by  this  formula,  yet 
there  are  cases  in  which  the  coefficients  (which  are  functions 
of  one  of  the  variables)  may,  by  giving  certain  values  to  the 
variable  they  contain,  become  infinite.  In  such  cases  the 
formula  cannot  be  applied;  for  in  general  such  values  for 


SUCCESSIVE    DIFFERENTIALS,  93 

that  variable,  would  not  reduce  the  function  itself  to  infin- 
ity, although  it  would   have  that  effect  on  its  development. 

Thus  in  the  function 

1 

we  have 

and  so  on. 

If  now  we  make  x^=^b^  all  the  coefficients  will  become  infi- 
nite, and  we  should  have 

u^=-a-^{b  —  xA;-yy^^^a-\-'^ 
by  the  formula  instead  of  having  as  we  ought 

which  cannot  be,  for  the  value  oi y  is  not  dependent  on 
that  of  .V,  and  hence  ?/  is  not  necessarily  infinite  when 
x^b;  but  for  all  other  values  of  x  the  formula  will  give  the 
true  development  of  the  function. 

And  herein  is  the  difference  between  the  formulas  of  Tay- 
lor and  Maclaurin ;  when  that  of  the  former  fails  it  is  for 
only  one  value  of  the  variable  ;  while  that  of  the  latter,  when 
it  fails  at  all,  fails  for  every  value  of  it. 

Note. — In  fact,  the  theorems  of  both  Taylor  and  Maclaurin  are  founded  on  the 
principle  illustrated  in  Art.  21.  The  real  object  of  both  is  to  find  from  the  rate  of 
change  of  a  function  what  will  be  its  new  state  arising  from  a  given  change  in  the  value 
of  the  variable. 

The  general  method  of  doing  this  is  to  find  the  successive  differentials  of  the  func- 
tion in  its  first  state,  and  then  to  multiply  the  successive  differential  coefficients  by  the 
successive  powers  (properly  divided)  of  the  actual  change  in  th6  variable  ;  this  will  give 
the  actual  successive  partial  changes  in  the  function  which  together  make  up  the 
entire  change,  and  thus  develop  the  function  in  the  new  state. 

For  this  purpose  the  variable  must  have  two  points  of  value;  one  where  the  func- 
tion is  to  be  (differentiated,  and  the  other,  the  new  value  produced  by  the  change  ;  and 
to  this  end  the  variable,  in  algebraic  functions,  is  made  to  consist  of  two  parts,  either 
by  making  it  a  binomial  or  something  that  may  be  reduced  to  that  form. 

In  Maclaurin's  theorem  the  variable  consists  of  a  constant  and  a  variable  combined 
together,  so  that  their  united  value  is  a  variable  one,  and  the  constant  part  is  simply 
one  point  in  that  variable  value.     This  is  the  point  at  which  the  differentiation  of  the 


94  DIFFERENTIAL    CALCULUS. 

function  is  made  ;  but  as  a  constant  cannot  be  dififerenliated  the  variable  is  attached 
to  it  long  enough  for  that  purpose  and  then  made  zero.  In  Taylor's  theorem  the  varia- 
ble is  the  sum  or  difference  of  two  others,  and  the  point  of  differentiation  is  when  the 
variable  has  reached  the  value  of  one  of  its  variable  parts.  This  being  a  variable,  the 
function  can  be  differentiated  directly,  and  the  other  variable  may  be  made  zero  before 
the  operation.  Hence  the  theorems Ipf  Maclaurin  and  Taylor  are  alike  in  this;  both 
have  a  compound  variable  having  two  points  of  value,  both  are  differentiated  at  the 
same  point,  and  the  successive  differential  coefficients,  which  are  precisely  alike  in  form 
and  value,  are  multiplied  by  the  successive  powers  of  the  change  in  value.  The  only 
difference  is  that  in  one  the  differentiation  at  the  required  point  is  ma.dQ  indirectly ^  and 
the  variable  change  made  zero  afterwards;  while  in  the  other  the  differential  is  made 
directly^  the  variable  change  being  made  zero  beforehand.  Hence  a  function  of  a 
binomial  variable  may  be  expanded  by  either  method.  By  Taylor's,  considering  both 
terms  variable  and  reducing  one  to  z^xo  before  differentiation  ;  or,  by  Maclaurin's,  by- 
considering  one  term  as  constant  and  reducing  the  other  to  zero  after  differentiation. 

Thus  in  the  case  of  the  function  \^~\~y]i  the  differential  coefficient  will  be  pre- 
cisely the  same  if  we  reduce  r  to  zero  and  differentiate  x  by  Taylor's  method,  or  con- 
sider jr  as  constant  and  reduce >/  to  zero  after  differentiation,  by  Maclaurin's  method. 

In  order  that  a  binomial  may  represent  a  single  variable,  both  terms  must  be  subject 
to  the  same  conditicns,  so  th  \\.  each  term  may  be  considered  as  a  part  of  the  same 
compound  variable  ;  and  the  failing  cases  in  Maclaurin's  and  Taylor's  methods  are 
simply  those  in  which  the  binomial  variable  becomes  a  monomial,  by  giving  the  variable 

J_ 
a  certain  value.     Thus  the  case  cited  in  Art.  24,     C -\-  aX"  ^   is  not  a  true   binomial 
variable,  since  the  terms  are  not  subjected  to  the  same  conditions.     If  we  make  it 

J_ 
\C~\~ClXj      we    have    a    true    binomial    variable,    and    the    differential    coefficient 

J. 

-^-f^-j-cX^)  will  not  reduce  to  infinity  when  x=o.     Similarly  the  case  cited  in  Art. 

JL 
27,  namely  U'^^Cl-\~\b  —  J\?-|-_)')     is  one  in  which  when    x-=b,  the  variable  in  the 

JL 
fUTCtion  reduces  to  y^  and  the  function  itself  to  ^-f-J^",  which   does  not  contain  a 
binomial  variable  of  the  required  form. 

The  same  principle  will  apply  to  transcendental  functions;  which,  in  order  to  be 
developed,  must  have  two  points  of  value  in  the  compound  variable— one  for  the  dif- 
ferential and  the  other  for  the  development.  Thus  Cl^k  may  be  expanded  by  Mac- 
laurin's theorem,  since  it  has  two  points  of  value,  one  at  k^  the  point  of  differentiation 
where  jr=c,  and  the  other  the  full  value  produced  by  x. 


SECTION    IV, 


MAXIMA  AND  MINIMA. 

(28)  We  have  seen  that  when  a  variable  changes  its  value 
at  a  uniform  rate,  the  value  of  its  function  will  in  general 
change  at  a  rate  that  is  not  uniform.  It  may  increase  at  a 
diminishing  rate,  until  at  a  certain  point  it  ceases  to  increase 
and  begins  to  diminish,  in  which  case  the  turning  point  is 
the  one  of  greatest  value,  and  is  called  a  maximum.  Or  it 
may  decrease  to  a  certain  point  and,  having  attained  its  min- 
imum, begin  to  increase.  The  problem  is  to  find  whether 
there  is  a  maximum  or  minimum  value  for  a  function,  while 
its  variable  is  uniformly  increasing,  and  if  there  is,  to  find 
the  corresponding  value  of  the  variable  and  of  its  function. 

(29)  While  a  positive  function  is  increasing  as  the  varia- 
ble increases,  its  rate  of  change  or  differential  will  be  posi- 
tive; and  negative  when  it  decreases  (Art.  3).  Hence  when 
a  function  is  passing  through  a  maximum  or  minimum  value, 
the  sign  of  the  differential  coefficient  must  change  from 
minus  to  plus  or  from  plus  to  minus  —  the  former  in  case  of 
a  minimum,  and  the  latter  in  case  of  a  maximum. 

But  sucli  a  change  can  take  place  only  while  the  differential 
coefficient  is  passing  through  zero  or  infinity.  Our  first 
inquiry  then  is  whether  there  is  any  finite  value  of  the  vari- 
able that  will  reduce  the  first  differential  coefficient  to  either 

95  % 


g6  DIFFERENTIAL     CALCULUS. 

of  these  values.  For  this  purpose  we  solve  the  equation 
formed  by  placing  the  first  differential  coefficient  equal  to 
zero,  and  thus  find  the  corresponding  value  of  the  variable. 
Here  we  have  one  of  three  results. 

First.  There  may  be  no  real  value  for  the  variable.  In 
this  case  there  is  neither  maximum  nor  minimum. 

Second.  There  may  be  a  real  finite  value  for  the  variable 
that  will  reduce  the  differential  coefficient  to  zero.  In  this 
case  there  will  pi-obably  be  a  maximum  or  minimum. 

Third.  There  may  be  no  finite  value  of  the  variable  that 
will  reduce  the  differential  coefficient  to  zcro^  but  at  the  same 
time  there  may  be  one  that  will  reduce  it  to  infinity.  In  this 
case  we  form  the  equation  by  placing  the  differential  coeffi- 
cient equal  to  infinity,  and  the  root  that  satisfies  the  equa- 
tion will  indicate  2^ probable  maximum  or  minimum. 

In  order  to  determine  in  the  two  latter  cases  whether  there 
is  a  maximum  or  minimum  value  of  the  function,  and  if  so, 
which  of  the  two  it  is,  we  may  substitute  in  the  function,  in 
place  of  the  variable,  a  quantity  a  little  less,  and  one  a  little 
greater  than  that  derived  from  the  equation.  If  the  result 
in  both  cases  is  less  than  when  the  true  value  is  substituted 
there  is  a  maximum;  if  greater,  there  is  a  minimum  value 
of  the  function  for  the  true  value  of  the  variable. 

We  may  also  determine  the  same  thing  by  substituting 
these  approximate  values  in  the  differential  coefficient,  which 
the  true  value  reduces  to  zero.  If  they  cause  the  result  to 
change  the  sign  from  plus  to  minus  by  substituting  first  the 
less,  and  then  the  greater  quantity,  there  is  a  maximum,  for 
the  function  is  passing  from  an  increasing  to  a  decreasing 
state.  If  the  change  is  from  minus  to  plus,  there  is  a  min- 
imum, for  the  function  is  passing  from  a  decreasing  to  an 
increasing  state  at  that  point. 


MAXIMA    AND    MINIMA. 


97 


EXAMPLE. 


Find  the  value  of  x  which  will  render  u  a  maximum  or 
minimum  in  the  equation, 


tc- 


■x^  —  gx^-\-24x  —  'j 


Differentiating    and    placing    the    differential    coefficient 
equal  to  zero  we  have 

f^=^x^-i8x-^24=s{x^--6x  +  8)=o 

from  which  we  find 

x^=4  and  x=^2 
If  we  substitute  in  the  function  and   in  the  differential 
coefficient  t,  2,  3,  4,  5,  etc.,  successively,  we   shall  have  for 


jr  =  i 

.    .  u=    g   . 

du_ 
•    •    dx  —  3 

X^2 

■  .  ?'^  =  i3  • 

dii. 

x=s 

.     .     .   ?/  =:  I  I     . 

x=4 

.   .  n=   g   . 

du 

^=5 

.  .   .  ?/  =  i3  . 

du 

•     ■   dx  —  3 

x=6 

.    .  u  =  2g  . 

du 
•     •    Sx  =  24 

Indicating  that  for  x  —  2  the  value  of  the  function  is  a 
maximum,  the  differential  coefficient  passing  from  plus  to 
minus;  and  for  x=4  the  value  of  the  function  is  a  mini- 
mum—the differential  coefficient  passing  from  minus  to  plus. 

(30)  It  must  be  understood  that  by  maximum  and  mini- 
mum is  not  meant  the  absolutely  greatest  or  least  value  of 
the  function,  but  the  /turning  pointy  from  an  increase  to  a 
decrease,  or  vice  versa.  Hence  there  may  be  as  many  max- 
ima or  minima  of  the  function  as  ther  ;  are  values  of  the 
variable  that  will  reduce  the  first  differential  coefficient  to 
zero  or  infinity. 

It  is  also  to  be  understood  that  in  the  discussion  of  this 
subject,  when  a  function  is  stated  to  be  an  imreasing  one,  it 


98  DIFFERENTIAL     CALCULUS. 

is  meant  that  it  is  either  positive  and  becoming  greater,  or 
negative  and  becoming  less.  If  it  is  said  to  be  decreasmg,  it 
is  either  positive  and  becoming  less  or  negative  and  becom- 
ing greater.     Thus  if  we  take  the  function 

u=ix'^  —  25 

and  make  x,  successively,  equal  to 

1.2.3.4.5.6.7. 
the  successive  values  of  the  function  will  be 
—  24,  — 21,— 16,  — 9,  o  +11, +  24 

and  it  is  said  to  be  increasing  throughout  the  whole  change^ 
although  at  first  its  numerical  negative  values  decrease.. 
This  is  also  indicated  by  the  sign  of  the  differential  coeffi- 
cient which  is  positive  as  long  as  x  is  positive.  So  that  the 
maximum  value  of  the  function  may  be  the  greatest  positive 
or  the  least  negative  value,  and  the  minimum  the  least  posi- 
tive or  the  greatest  negative  value. 

(3 1 )  There  is  another  method  of  ascertaining  whether  the 
first  differential  coefficient  changes  its  sign  on  passing 
through  zero  or  infinity,  for  this  is  the  unfailing  test  of  a 
maximum  or  minimum.  Having  found  that  value  of  the 
variable  which  reduces  the  first  differential  coefficient  ta 
zero,  substitute  that  value  in  the  second  differential  coeffi- 
cient ,  if  it  contain  the  variable;  then 

First.  If  it  reduces  the  second  differential  coefficient  to 
a  negative  quantity,  it  indicat,is  that  when  the  first  is  at  zera 
it  must  be  a  decreasing  function,  which  can  be  at  that  point 
only  by  its  passing  fro7n  a  positive  to  a  negative  state,  and 
hence  the  function  itself  must  be  passing  from  a  state  of 
increase  to  a  state  of  decrease,  and  hence  is  at  a  maximum. 

Second.  If  it  reduces  the  second  differential  coefficient  to 
a  positive  quantity,  it  indicates  that  the  first  when  at  zero  is 
an  increasing  function,  and  must,  therefore,  be  passing /r^;// 
a  negative  to  a  positive  state,  hQnce  the  function  is  passing  from. 


MAXIMA    AND    MINIMA.  99 

a  decreasing  to   an  increasing  state,  and  is,  therefore,  at  a 
minimum. 

Third.  If  it  reduces  the  second  differential  coefficient  to 
zero,  we  may  resort  to  the  third ;  and  if  the  same  value  of 
the  variable  reduces  that  to  a  real  finite  quantity,  either  pos- 
itive or  negative,  it  shows  that  the  second,  at  zero,  is  chang- 
ing its  sign,  and,  therefore,  the  first  is  changing  from  an 
increasing  to  a  diminishing  function,  or  vice  versa,  and, 
therefore,  does  not  at  the  zero  point  chatige  its  sign.  Hence 
there  is  neither  maximum  nor  minimum  in  the  value  of  the 
function. 

Fourth.  If  it  reduces  the /////'^/differential  coefficient  to 
zero  we  may  resort  to  X\\q  fourth.  If  it  reduces  this  to  a  real 
finite  value,  it  indicates  that  at  zero  the  third  changes  its 
sign,  for  it  can  increase  on  both  sides  of  zero  only  by  pass- 
ing from  negative  to  positive,  or  diminish  on  both  sides  by 
passing  from  positive  to  negative.  This  will  show  that  the 
second  coefficient  does  not  change  its  sign,  for  if  it  increases 
befor.e  it  becomes  zero  and  decreases  afterwards,  or  vice  versa^ 
it  can  approach  the  zero  point  only  until  it  touches  it,  and  then 
must  recede  without  changing  its  sign.  This  proves  that  the 
first  coefficient  does  change  its  sign,  for  since  the  second  does 
not  change  the  first  must  be  passing  through  from  one  side 
to  the  other.  There  will,  therefore,  be  a  maximum  or  mini- 
mum— the  first  if  the  fourth  differential  coefficient  has  a 
negative  value,  and  the  second  if  it  is  positive. 

We  may  continue  thus  and  show  that  if  the  first  differen- 
tial coefficient  that  is  reduced  to  a  real  value,  by  substituting 
that  value  of  the  variable  that  reduces  the  first  to  zero,  is  of 
an  eve?i  order  and  positive^  there  will  be  a  viinimum;  if  it  is 
7iegative,  there  will  be  a  maximum;  and  if  it  is  of  an  odd 
order  there  will  be  neither  maximum  nor  7?iini?nuni. 

Fifth.  If  any  value  of  the  variable  reduces  the  first  dif- 
ferential coefficient  to  infinity,  it  w  11  probably  reduce  all  the 


lOO  DIFFERENTIAL    CALCULUS. 

succeeding  ones  to  infinity,  also.  It  will,  therefore,  be  best 
in  such  a  case  to  substitute  values  for  the  variable  a  little 
less  and  then  a  little  greater  than  the  one  found.  If  the 
value  of  the  first  differential  coefficient  changes  from  plus  to 
minus  there  is  a  maximum,  and  the  second  will  be  plus  on 
both  sides  of  infinity  ;  for  the  first  must  be  an  increasing  pos- 
itive function  in  order  to  become  positively  infinite,  and  if 
negative  on  the  other  side,  must  be  a  decreasing  function, 
for  it  cannot  be  an  increasing  negative  function  on  leaving 
infinity.  Hence  (Art.  3)  the  second  must  be  positive  in  both 
cases. 

Sixth.  If  the  first  differential  coefficient  in  the  last  case 
changes  from  minus  to  plus  there  will  lea  minimum,  and 
the  second  coefficient  will  be  minus  on  both  sides  of  infinity. 
Thus  we  see  that  when  any  value  of  the  variable  reduces 
the  first  differential  coefficient  to  zero,  and  is  substituted 
in  the  second,  a  minus  result  indicates  a  viaxinmm  in  the 
function,  and  a  phis  result  a  minimum.  When  any  value 
reduces  the  first  coefficient  to  infinity.,  2^  plus  sign  for  the 
second  indicates  a  maximtun,  and  a  minus  sign  a  minimum. 

EXAMPLES. 

Ex.  I.  In  order  to  illustrate  the  first  case  in  this  article 
we  take  the  function 

u'=\(ix—x'^  (i) 

from  which  we  obtain  by  differentiation 

— =16  —  2^  (2) 

cix 

d^u  f  •, 

We  find  that  jv=8  will  reduce  the  first  differential  coefficient 
to  zero,  while  the  sign  of  the  second  is  minus.  Hence  at 
x=-^  the  first  must  be  a  decreasing  function,  and,  therefore, 


MAXIMA    AND    MINIMA. 


lOI 


passing  through  zero  from  plus  to  minus,  the  function  will, 
therefore,  be  an  increasing  one  to  that  point  and  then  a 
diminishing  one;   hence  a  maximum. 

If  we  substitute  in  the  function  and  the  first  differential 
coefficient  for  x  values  a  little  less  and  a  little  greater  than 
8,  we  have  for 


du 

dx  —  2 


X- 


=  6- 


dx 


=o 

du 


■9  .  .  .  .  ?/=03  ....  di=  — 2 

If  we  represent  the  values  of  the  function  by  the  ordi- 
nates  of  the  curve  ABC  (Fig.  3),  the  curve  itself  will  cor- 
respond to  the  range  or  locus  of  values  of  the  function,  while 
the  variable  increases  uniformly  in  passing  from  7  to.  9. 
From  A  to  B  the  function  increases, 
but  at  a  decreasing  rate,  and  hence 
the  first  differential  coefficient  is 
positive  but  decreasing  until  it 
reaches  zero  at  B.  The  function  then 
decreases  at  an  increasing  rate,  and 
hence  the  first  differential  coefficient 
must    be    negative    and    increasing. 

But  when  this  coefficient  (or  any  variable)  is  positive  and 
decreasing,  or  negative  and  increasing,  its  rate  of  change,  i.e.  ^ 
the  second  differential  of  the  function,  must  (Art.  3)  be  neg- 
ative throughout,  which  corresponds  with  the  result  found  in 
equation  (3). 

Ex.  2.     T©  illustrate  the  second  case  we  take 
u^x'^  —  iGjf-f-yo 
from  which 


(i) 


dn 

-7— =  2.r—  16 

ax 

d^  If 


dx" 


=2 


(2) 
(3) 


I02 


DIFFERENTIAL    CALCULUS. 


clu 


We  infer  from  equation  (3)  that  j^  is  an  increasing  func- 
tion for  all  values  of  Xy  and  hence  it  is  so  when  ^=8,  which 
reduces  j^  to  zero.  From  which  we  infer  that  j^  is  passing 
from  a  negative  to  a  positive  state,  and  the  function  itself 
from  a  decrease  to  an  increase.      Hence  a  minimum. 

If  we  substitute  7,  8  and  9  successively  for  x  in  equations 
(i)  and  (2),  we  have  for 

?/=7 

?^— 6 


X- 


X- 


x^=g     71^=^'] 


clx  2 

(III 

dx—O 

(hi 
dx' 


==2 


4- 


which  corresponds  with  our  deductions. 

If  we  let  the  ordinates  of  the  curve  ABC  (Fig.  4)  rep- 
resent the  values  of  ?/,  we  see  that  from 
A  to  B  the  value  of  ?/  diminishes,  as  is 

shown  by  the  sign  of  ^,  and  at  a  dimin- 
ishing rate  as  is  shown  by   the  positive 

sign  of -^-y  (Art.  3).     From   B   to   C   2^ 

increases,  as  is  shown  by  the  positive  sign  of  ^^!,  and  at  an 

increasing  rate,  as  is  shown  by  the  sign  of  —~i ,    which    is 

still  plus.     Hence  the  shape  of  the  curve. 
Ex.  3. 

whence 


du 


To  illustrate  the  third  case  we  make 
?^ =9 -1-2(^—3)^ 


du  _ 
dx 
d"^  u 


=6(^-3)2 
^^.=12(^-3) 


(i) 

(2) 

(3) 


du 


Here  we  find  :r  — 3  reduces  -^x  to  zero,  and   hence  if  there 
IS  a  maximum  or  minimum   it  will   be  for  that   value  of  x. 


.MAXIMA    AND    MINIMA. 


103 


—        .       .  .  d  ~  II  . 

But  It  also  reduces     ,  ..,    to  zero,  hence  we  resort  to  the 

dx" 

d  ^  u        . 
value  of     ,    ■ ,  which  we  find  to  be  12.     We  infer  from  this 


that  when 


dx 
d  ~  21 


dx'''' 


'■o  it  is  passing  from  negative  to  positive, 


flit 


hence  ^is  passing  from  a  decreasing  to  an  increasing  func- 
tion at  the  zero  point,  and,  therefore,  does  not  pass  through 
it.  It  is,  therefore,  all  the  time  positive,  and  the  function  is 
at  all  times  an  increasing  one,  so  that  there  is  neither  max- 
imum nor  minimum.  We  may,  also,  learn  the  same  thing 
from  inspection,  for  since  the  value  of  ^i  is  a  squar'e  it  must 
always  be  positive. 

If  we  substitute  in  the  given  function  and   in  the  values 

,  dll  ,    d'^  7i    ,  ,  ,  ... 

of   7-  and  —TT  the  numbers  2,  3  and  4  successively  for  x, 
dx  dx- 

we  have  for 


JC  =  2 

x=2, 
x^=^^ 


di( 
dx 
d_u^ 
dx 
du 


=6 


=0 


tc- 


II   ^.-=6 

dx 


d  ~  71 

— -7=-I2 
dx- 

d  -  // 

~d^ 

d'  u 


=■0 


dx'^ 


12 


If  we  let  the  ordinates  of  the  curve  ABC  (Fig.  5)  repre- 
sent the  values  of  u^  we  see  that  from  2  to  3  the  function 
increases,  as  is  shown  by  the  positive 
value  of  ^,  but  at  a  decreasing  rate, 
as  is  indicated  by  the  negative  sign  of 

7  o 

,  ^    (Art.   3),  between  those  points 

or  while  x  is  less  than  3.     From  3  to 
4  the  function  is  still  increasing,  as  is 


104  DIFFERENTIAL    CALCULUS. 

shown  by  the  positive  sign  of  ^7^,  and  at  an  increasing  rate^ 

ci  '"  21 

as  is  shown  by  the  positive  sign  of     ,  .,   when  the  value  of 

ax" 

X  is  greater  than  3.     Hence  at  B  where  the  value  of  ^  is 
zero,  the  function  having  increased  at  a  decreasing  rate  up 
to  that  point,   ceases    for    an    inappreciable    moment,  and 
begins  again  to  increase  at  an  increasing  rate. 
Ex.  4.     To  illustrate  the  fourth  case  take 

2^=5+(^-7)4  (i) 

whence 

;^=4(^'^-7)^  (2) 

-^  =  12(^^-7)^  (3) 


dx 
d  ^  u 


dx^ 
d  ^  u 
dx^ 


=  24(^^-7)  (4) 


^24  (5) 


Here  we  see  that  the  fourth  differential  coefficient  is  the 
first  that  has  a  real  finite  value  for  x=:7,  which  reduces  all 
the  preceding  ones  to  zero.  Hence,  according  to  our  rule, 
there  should  be  a  minimum  value  for  the  function  at  that 
point.  In  fact,  the  sign  of  this  coefficient  shows  that  the 
third  changes  at  zero  from  minus  to  plus  ;  and  this  shows 
that  the  second  does  7iot  change  its  sign  at  zero,  but  after 
being  a  decreasing  function  to  that  point,  becomes  an  in- 
creasing one,  and  is,  therefore,  positive  both  before  and  after. 
And  this  again  shows  that  the  first  does  change  its  sign  at 
zero,  since  it  is  an  increasing  function  on  both  sides  of  zero, 
which  can  be  only  by  passing  from  a  negative  to  a  positive 
value.  Hence  the  function  will  decrease  until  :^=7,  after 
which  it  will  increase,  showing  a  minimum  at  that  point. 

If  we  substitute  in  the  given  function  and  in  the  differen- 


MAXIMA    AND    MINIMA.  IC5 

tial  coefficients  the  numbers  5,  6   7,  8,  9  successively  for  ^ 
we  shall  have  for 

du  d"  u       ^     d"^  u 


•^■-5 

u  —  21 

^/;c 

3^ 

^;c- 

-46 

^/^•^ 

--46 

.r  =  6 

?^=   6 

ti  

-4 

(( 

=  12 

i( 

=  -24 

.T  =  7 

u=   5 

*'    =::     0 

a 

—     0 

(( 

=     0 

.r=8 

z/=   6 

''  -   4 

u 

=  12 

u 

=  24 

.T^^9 

Z/  =  2  I 

"  -32 

u 

=  48 

(( 

=  48 

which  illustrates  the  conclusions  we  have  drawn. 

The  general  proposition  enunciated  in  the  fjuurth  case  may 
be  demonstrated  analytically  as  follows.     Let  us  suppose 

u  =  F[x) 
and  let  the  variable  x  be  first  increased  and  then  diminished 
by   another  variable  h  j   and   let   these   new  states  of  the 
function  be  represented  by  u   and  u\  then  we  have 

u  —F{x-\-/i) 

7i"  =^F(x—/i) 
Developing  these  by  Taylor's  theorem  we  have,  after  sub 
tracting  the  original  function  u, 

,        —^^^  7,\^~^^     h-     d"^  u     h"^ 

dx  ''  '  dx'''' '     2       dx^  '2.3' 

,/  du  ,  .  {F  u     h^     d^  u     //2 

u  —w^ — ;~^^+^r^ 7~-j  • i"   etc. 

dx        dx~       2       dx-^      2.3 

Since  the  powers  of  /i  increase  in  each  successive  term  of 
this  development,  we  may  reduce  the  value  of /^  to  such  an 
extent  that  the  value  of  any  one  term  shall  be  greater  than 
that  of  all  the  succeeding  terms  added  together.  Such  in 
fact  will  be  the  case  if  /i  is  less  than  one-half  m  the  series 
/?,  /^*,  /i^,  etc. 

Let  us  suppose  /i  to  be  so  reduced,  then  if  7i  is  a  maxi- 
mum, it  must  be  greater  than  u'  or  u",  and  the  second  mem- 
bers of  both  these  equations  must  be  negative;  if  it  is  a 
minimum  it  must  be  less  than  u  or  u",  and  in  this  rg.se  the 
second    members    of    both    equations    must    be    f^^itive. 


Io6  DIFFERENTIAL    CALCULUS. 

Hence,  in  case  of  a  maximum  or  minimum,  the  second 
members  of  both  equations  must  have  the  same  sign,  and 
the  first  term  of  each  (which  controls  the  value  of  all  the  rest) 
having  contrary  signs  must  reduce  to  zero ;  that  is,  ^-^  must 
be  zero,  since  /i  is  not.  This  then  is  a  necessary  condition 
to  a  maximum  or  minimum.     If  there  is  a  real  value  for  the 

second  term  in   each   equation,   its   sicjn    (or  that   of  -^-^ 

^  ax" 

since  /i"  is  positive),  will  now  control  that  of  the  whole  sec- 
ond member,  and  will  determine  whether  ^^  is  a  maximum  or 

minimum.     If  the  second  term  (or  —r-ir]  become  zero,  there 

ax-  / 

can  be  no  maximum  or  minimum   unless  the  third  term  (or 

— ,- )  which  is  now  the  controUmg  term,  is  also  zero,  since 
ax-^  ' 

it  has  contrary  signs  in  the  two  equations.  We  see  then 
that  the  conditions  of  a  maximum  or  minimum  are:  first, 
that  the  first  differential  coefficient  should  become  zero; 
and,  second,  that  the  first  succeeding  differential  coefficient 
that  has  a  real  value  should  be  one  of  an  even  order,  since  the 
even  terms  have  the  same  sign  in  both  equations.  If  that 
is  negative,  the  whole  of  the  second  member  of  the  equa- 
tion is  negative,  and  there  is  a  maximum ;  if  it  is  positive, 
there  is  a  minimum.  Which  agrees  with  the  rule  already 
found 

Ex.  5.     To  illustrate  the  fifth  case  we  take 


whence 


'■~\o 

-{x- 

-3)' 

du 

— '. 

7 

dx~ 

3(-v- 

■3)* 

d^  u 

2 

dx^ 

9(.v 

-3)* 

Z/  — 10  — (.T— 3)=^  (l) 

(3) 


MAXIMA    AND    MINIMA. 


107 


dtt 
Here  we  find  that  x^^^  will  reduce -t— to  infinity.     Refer- 

ing  to  the  value  of     ,    ,    we  find  that  it  reduces  that  also  to 

infinity,  but  we   see  by  inspection  that   any  other  value  for 

d  ^^  u 


X,  whether  greater  or  less  than  3,   will  make  that  of 


dx'-^ 


dti 


positive.     We  see  also  that -^  will  be  positive  when  .r  is  less 

than  3,  and  negative  when  it  is  greater.  From  all  this  we 
infer  that  the  function  is  an  increasing  one  before  ^'  =  3,  and 
a  decreasing  one  afterwards.  Hence  there  is  a  maximum 
at  that  point. 

If  we   substitute   for  jt,  in  equations   (i),  (2)  and  (3),  the 
numbers  2.3.4  successively,  we  have  for 


dii      2 
^    dx       -^ 
du 

X^=-l        ?/  =  I  O     =  CO 

'^  dx 


d  ~  u 


du 
x=A     ?^=    g  —7-  = 

i/x 


dx" 
d^  u 
dx^ 

^    dx-  ■ 


:00 


If  we  let  the  ordinates  of  the  curve   ABC  represent 
the  successive  values  of  u  (Fig  6.)  we 
see  that  from  2  to  3  the  function  in- 
creases, as  is  shown  by  the  positive 

value  of -^—  and  at  an  increasing;  rate 

dx  ^ 

as  is  shown  by  the  positive  value  of 

—r-r.     From  "^  to  4  the  necrative  sisfn 
dx^  v)       -+  »  o 

of -y- indicates  a  decrease  of  the  function,  while  the  positive 

sign  of       ^  (which  has  not  changed)  shows  that  this  decrease 

is  at  a  decreasing  rate.     Hence  the  form  of  the  curve. 
£x.  6.     To  illustrate  the  sixth  case  we  take 


io8 


DIFFERENTIAL    CALCULUS. 


whence 


2/  =  (3Jc— 9)3 

du  _        2 
dx 


d^  u 


(3^-9)3 


dx'' 


(3^-9)3 


(i) 

(2) 
(3) 


du 


In  this  case,  as  before,  we  find  that  x'='i  reduces  -7-  and 

d    u  flu 

—TY  ^°  infinity.     We  see  also  by  inspection  that  — -  changes 

CLX  (IX 

from  minus  to  plus  as  x  passes  from  less  than  3  to  greater, 

d    u 
while   ,  „    is  negative  on  both  sides  of  infinitv.     Hence  we 

dx" 

infer  that  ?/  is  a  decreasing  function  until  ^-=3,  and  an  in- 
creasing one  afterwards.     Hence  a  minimum  at  that  point. 
If  we  substitute  for  x  in  equations  (i),  (2),  (3),  the  num- 
bers 2.3.4  successively  we  shall  have  for 

,        du  2      d  ^  ti  2 


x- 


u- 


Jt^=3     u^=-o 


X- 


3  ,_ 

•4      U  —  <sJ 


dx 
du 

=  00 
2 

d^"             1/ 

V  81 

d"  u 

dx 
du 

d^  u             2 

dx 

dx-                  3/    g^ 

If  we  let   th©  ordinates  of  the  curve  ABC  (Fig.  7)  rep- 
resent   the    successive    values    of    u 
we  see  that  from  2   to  3  the  function 
decreases,  as  is  shown  by  the  nega- 

tive  value  01  —— ,  and  at  an  increas- 

dx  ' 

ing  rate,  as  is  shown  by  the  negative 

d  ^  u 
value  of  -^^;  from  3  to  4  the  pos- 


dx' 


du  . 


itive  value  of -7- indicates  an  increasing  function,  while  the 


MAXIMA     AND    MINIMA 


109 


negative  value  of  ——7  shows  that  increase  to  be  at  a  dimin- 

ishing  rate.     Hence  the  form  of  the  curve. 

(32)  We  have  seen  (Art.  30)  that  there  may  be  as  many 
maxima  and  minima  of  a  function  as  there  are  roots  for  the 
•equation  formed  by  making  the  value  of  1^—^.     To  illustrate 

a  case  of  this  kind  we  take 
Ex.  7. 

tl^=X^  —  2QX'^-\-\'^~^2X~—2)20X-\-2%()  (l) 

du 

X 


whence 


—4:r^—6oA;'-^4- 264:^  —  320  (2) 


dx 


=  I2.X'  —  \20X-\-26^  (3) 


Placing  the  second  member  of  equation  (2)  equal  to  zero, 
we  find  for  x  three  values  as  follows : 


X^^2 

x  =  ^ 


Substituting  these  values  in  equation  (3)  we  have  for 

d^^  u 
x=^2    —-^=72 
dx- 


d"^  '/ 
dx^ 

d  -'  u 
~d^ 


^=5    -7Z^  =  -3^ 


x  =  8 


from  which  we  infer  that  for  x  =  2  and  x=S  there  is  a  mini- 
mum, and  for  .t=5  there  is  a  maximum.  This  will  be  seen 
by  substituting  for  x  in  equations  (i),  (2),  (3),  successive 
values,  as  follows : 


no 


DIFFERENTIAL    CALCULUS. 


X^=-    o 

?/  =  286 

-  = 

-320 

dx 

u 
—  =  264. 

X=     I 

u=   79 

— 

—  112 

=  156 

X^=     2 

z/=  30 

= 

0 

=    72 

x=   3 

x=   4 

x=   5 

^^=  si 

2r-=   94 

Z/  =  I  I  I 

= 

40 
0 

=    12 
=  —  24 
=  -36 

x^=^   6 

?/=  94 
^^=  55 

=. 

-  32 

-  40 

==  —  24 
=    12 

j*:=  8 

2/=  30 

zzz 

0 

=   72 

X=IO 

u=   79 

?/  — 286 

zm. 

112 

320 

=  156 
=  264 

If  we  let  the  ordinates   of  the  curve  ABC  (Fig.  8)  rep- 


resent the  successive  values  of  t/-  corresponding  to  numbers 

substituted  for  x  at  the  foot  of  each,  we  see  that  the  function 

decreases  at  a  diminishing  rate  until  jv  =  2,  when  it  ceases  to 

decrease  and  begins  to  increase  at  an  increasing  rate,  as  is 

.       .   du 
shown  by  the  change  of  sign  my-  and   the   continued  posi- 

d^ti 
tive  sign  of  y^.     But  at  x^=4,  although  still  increasing,  as 

...  du    ,     , 

IS  shown  by  the  posLive  sign  of  y-,  it    is    at    a  diminishing 


ix 


d^-u 


rate,  for    ,  2  is  now  negative,  and  thus  continues  until  at  5, 

du  .       .         . 

-7-  becomes   zero,  the  function  ceases  to   increase  and  be- 

dx 

gins  to  diminish  at   an   increasing   rate,  as  is  shown  by  the 

.  du  d^  u  .        ^ 

negative  signs  of —- and -7^  at  ^=6.       But  at  jf=7  we 


dx 


dx"" 


MAXIMA    AND    MINIMA.  Ill 

(i    u  ... 

have  ^   positive,  showing  that  the  rate  of  diminution  of 

the  function  has  ceased  to  increase  and  begins  to  diminish, 
until  at  x  —  Z  it  has  become  zero,  when  the.  changes  at  x  —  2 
are  repeated.     We  notice  that  between  :r  =  3  and  x—^^  the 

d^  u 
sign  of  -7-^  changes  from  plus  to  minus,  showing  that  be- 
tween those  two  points  the  rate  of  —  has  changed  from  an 

increase  to  a  decrease,  that  is,  the   function  has  changed 

from  increasing  at    an    increasing    rate  to  increasing  at  a 

diminishing  rate.     The  exact  point  where  this  change  takes 

<^  2  ?/ 
place  is  where  the  value  of  — ^-;^=o.     This  will  give 

^=5-V  3  and  ^-5+^/3 

which  last  value  corresponds  to  a  point  between  x^()  and 
•^—7)  where  the  same  change  is  repeated,  only  in  a  contrary 
direction. 

From  all  these  cases  we  deduce  the  following  rule  for 
ascertaining  the  values  of  the  variable  that  will  produce  a 
maximum  or  minimum  value  for  the  function,  if  there  be 
any. 

Place  the  first  differential  coefficient  equal  to  zero ;  and 
substitute  each  of  the  roots  of  this  equation  for  the  varia- 
ble in  the  second  differential  coefficient.  Each  one  that 
reduces  it  to  a  real  negative  quantity  will  produce  a  maxi- 
mum value  for  the  function  ;  while  a  similar  positive  result 
will  indicate  a  minimum.  Should  any  real  root  thus  found 
reduce  the  second  differential  coefficient  to  zero,  substitute 
it  in  the  third,  fourth,  etc.,  successively,  until  a  real  finite 
value  is  found  for  some  one  of  the  coefficients.  If  the  first 
thus  found  be  of  an  even  order  and  positive,  there  will  be  a 
minimum  ;  if  negative  there  will  be  a  maximum.  If  the  first 
that  is  reduced  to  a  real  finite  quantity  is  of  an  odd  order, 


112  DIFFERENTIAL    CALCULUS. 

whether  positive  or  negative,  there  will  be  neither  maximum 
nor  minimum. 

The  first  differential  coefficient  may  also  be  placed  equal 
to  infinity,  and  if  there  be  any  real  finite  roots,  they  may  be 
treated  in  the  same  manner  as  those  obtained  by  placing  it 
equal  to  zero.  In  this  case,  however,  a  positive  sign  of  the 
second  differential  coefficient  indicates  a  maximum  and  a 
negative  sign  a  minimum. 

If  a  given  function  contains  two  variables  there  must  be 
an  equation,  and  one  of  the  variables  must  be  considered  as 
dependent  on  the  other.  The  problem  will'  be  to  find  the 
maximum  or  minimum  value  of  the  dependent  variable;  for 
which  purpose  it  must  be  considered  as  an  implicit  function 
of  the  other,  and  the  differential  coefficients  will  be  found 
as  in  other  cases. 

Note. — It  maybe  objected  that,  herein,  the  subject  of  maxima  and  minima  has 
been  treated  in  too  prolix  a  manner,  and  the  reasoning  has  been  unnecessarily  repeated. 
I  reply,  it  is  of  the  highest  importance  that  the  student  should  have  not  only  a  clear 
and  correct,  but  2,fa7niliar^  conception  of  the  laws  which  govern  the  relations  of  the 
different  orders  of  rates  or  differentials,  because  these  are  among  the  fundamental  ideas 
of  the  ca'culus,  and  essential  to  a  complete  comprehension  of  the  subject.  Now  unless 
these  ideas  are  presented  sufficiently  often  to  render  them  familiar;  if  the  student  on 
every  new  occasion  is  obliged  to  draw  afresh  upon  his  powers  of  imagination,  and  go 
through  the  mental  labor  of  forming  his  conceptions  anew,  the  study  will  prove  not 
only  more  difficult,  but  far  less  attractive.  He  will  be  like  a  traveler  in  the  dark,  who, 
instead  of  carrying  a  constantly  shining  lamp  to  guide  his  footsteps,  must  light  his 
candle  anew  for  every  fresh  obstacle.  Hence  the  importance  of  a  full  and  elaborate 
explanation,  even  at  the  expense  of  some,  otherwise  unnecessary,  repetition. 

EXAMPLES 

Find  the  value  of  x  for  the  maximum  or  minimum  value 
•of  u  in  the  following  equations : 

Ex.    8.     ?/=.T^-|-i8.v* +  io5>r.  Ans. 

Ex.     9.     iL'=^a  —  bx-\-x'^.  Ans 

Ex.  10.     2i-'=^a^-\-b'^  X — cx'^ .  Ans. 

Ex.  II.     //  =  3(^'-^:<::^— <^^jc+^^.  A?is. 

Ex.  12.     ti^=^a^  -\-bx'^  —cx"^ .  Ans. 


MAXIMA    AND    MINIMA.  II3 

APPLICATION    TO    PRACTICAL    PROBLEMS. 

(33)  In  order  to  apply  the  rules  for  determining  maxima 
and  minima  of  functions  to  the  solution  of  practical  prob- 
lems, it  is  necessary  to  obtain  an  algebraic  expression  of  the 
function,  whose  maximum  or  minimum  is  to  be  determined, 
in  such  terms  that  it  shall  contain  but  one  variable.  No 
specific  rules  can  be  given  for  this  purpose,  but  care  must 
be  taken  to  express  the  function  in  terms  of  a  variable  that 
shall  have  a  range  of  values  beyond  that  which  may  be 
required  to  produce  a  maximum  or  minimum,  for  if  it  does 
not,  although  there  may  be  a  kind  of  maximum  or  minimum, 
it  will  not  be  one  in  the  meaning  of  the  terfn  as  used  in  the 
calculus,  as  there  will  be  no  turning  poifit  in  the  value  of  the 
function,  nor  any  change  of  sign  in  the  value  of  the  first 
differential  coefficient.  A  few  examples  will  indicate  the 
nature  of  the  process  more  clearly. 

Ex.  I.  Divide  the  quantity  a  into  two  such  parts  that 
their  product  shall  be  a  maximum. 

Let  X  be  one  of  these   parts,  then  the  other  will  be  a—x, 

and  the  function  will  be 

x{a  —  x)  =^izx  —  x^ 

which  is  to  be  a  maximum.      Representing  it  by  u  we  have 

d//     _  .   . 

■a  —  2X  (i) 


(/X 

d~  u 


dx~ 

^.     .         .  .  .  du  .  , 

Placing  the  value  of  ^—  equal  to  zero  we  have 
°  dx 


(-') 


which  the  negative  sign  of  -j-^r  shows    to    be   a    maximum. 


d'^  u 
d^ 

Hence  w/ien  a  quantity  is  divided  into  two  parts  their  product  is 
a  maximum  when  they  arc  equal. 
8 


114 


DIFFERENTIAL    CALCULUS. 


Ex.  2.     To  find  the  greatest  cylinder  that  can  be  inscribed 
in  a  given  right  cone. 

Let  the  height  SC  (Fig.  9) 
of  the  cone  be  represented  by 
a.  and  the  radius  of  the  base 
AC  by  b,  and  let  x  represent  the 
distance  SD  from  the  vertex  of 
the  cone  to  the  upper  base  of 
the  cylinder.  From  the  trian- 
gles SAC  and  SED  we  have 
SC  :  AC  : :  SD  :  ED  or  a  :  ^  : :  ^  : 
ED,  hence 

But  the  area  of  the  upper 
base  of  the  cylinder  is 


Fig.  9. 


7:ED"  =- 


b^x^ 


Multiplying  this  by  DC=a—x,  the  height  of  the  cylinder, 
we  have  the  volume  or  capacity,  which  we  will  represent  by 
V,  and  hence 

V——^x'^(a—x) 

xTow  any  value  of  :r  that  will  render  x'^{a—x^  a  maximum 

will  render  any  multiple  of  it   also   a   maximum,  hence  — 5- 

a 

being  a  constant  factor  may  be  disregarded  in  the  operation. 

Differentiating  twice   and   representing  x^{a  —  x)   by  u^  we 

have 

^u 

dx 


^■'zax—'^x'' 
^2a — 6x 


dx" 

Making  the  value  of  -^x  equal  to  zero  we  have 

x'=o  and  ^=-0 


MAXIMA    AND    MINIMA.  II5 

The  first  cannot  be  a  maximum  since  it  reduces  the  value  of 

d'^  u 

-—T-  to  2a,  which  being  positive  indicates  a  mmmium.     In 

fact,  when  x=^o  the  cylinder  is  reduced  to  the  axis  of  the 
cone,  and    vanishes   with    x.      The  other  value   x=^a  will 

solve  the  problem,  since  it  reduces  the  value  of  -r-j-  to   —la 

a  negative  quantity  which  indicates  a  maximum  value  for  the 
function.     Hence 

TAe  maxifnuin  cylinde?-  thai  can  be  inscribed  in  a  right  cone  is 
one  in  which  the  height  of  the  cylinder  is  one -third  of  the  height 
of  the  cone.  The  radius  of  the  base  will  also  be  equal  to 
two- thirds  that  of  the  base  of  the  cone.  The  volume  of  the 
cylinder  will  be  to  that  of  the  cone  in  the  ratio  of  their 
bases,  or  as  4  is  to  9. 

Ex.  3.  Required  to  determine  the  dimensions  of  a  c^-lin- 
drical  vase,  that  will  contain  a  given  quantity  of  water  witli 
the  least  amount  of  surface  in  contact  with  it. 

Let  V  represent  the  given  volume  of  water,  x  the  radius 
of  the  base  of  the  cylindrical  vase,  and_y  its  altitude.  Then 
we  shall  have 


v^=^~x'^y 


from  which 


}---. 


■■x" 

Now  the  convex  surface  of  the  cylinder  is  equal  to  2-xyy 
and  substituting  the  value  oi y  we  shall  have  the  convex  sur- 
face equal  to 

2r.xv  _  2V 
-A'-  X 

If  to  this  we  add  the  surface  of  the  base  =-'X'  we  have  the 
whole  surface  in  contact  with  the  water.  Calling  this  surface. 
S  we  have 


ii6 


DIFFERENTIAL    CALCULUS. 


X 


^S 


2V 


X' 


•J-2TZX 


dx^ 


47>    . 


(0 

(2) 

(3) 


dS 


Placing  the  value  of  ax  equal  to  zero  we  have 


whence 


27:^^  =  27^ 


3/ 


This  value  answers  to  a  minimum  since  it  renders  the  value 


.of 


dx 


-  positive.     If  we   substitute   this   value  of  x  in  the 


expression  we  found  for  the  value  of  j;,  namely 

V 

y- 


TTX' 


we  have 


y'' 


^^5 


"hence  f^e  mim'mum  su?-face  7vill  be  m  contact  7vith  the  water 
when  the  height  of  the  cylinder  is  equal  to  the  radiiis  of  the  base. 

Ex.  4.     It  is  required  to  inscribe  in  a  sphere  a  cone  which 
shall  have  the  greatest  convex  surface. 

Suppose  the  semi-circumference  AMB  (Fig.  10)  to  revolve 
about  the  axis  AB,  it  will  describe 
the  surface  of  a  sphere,  and  the 
chord  AM  will  describe  the  con- 
vex surface  of  a  right  cone  in- 
scribed in  the  sphere;  AP  will 
be  its  height,  and  PM  the  radius 
of  its  base.  The  convex  surface 
of  the  cone,  which  we  will  call  S, 
will  be  F's-  ^°- 


MAXIMA    AND    MINIMA.  II7 

S  =  2t:PM  .  iAM=7rPM  .  AM  (r) 

We  have  now  to  determine  PM  or  AM,  either  of  which 
will  determine  the  other 
Let 

AB  —  2a  and  AP=^ 

then 
and 

Again 

AM=\/2ax 
Substituting  these  values  in  equation  (i)  we  have 


PM    =AV  .  ?B=x{2a—x) 


FM=\^x{2a—x) 


(2) 


S=7:PM  .  AM=-!:^  2ax—x2  .  '\/ 2ax=-^'\/ 4a2x-2—2ax3 
Differentiating  this  function  we  have 

</S       4.a^x—^ax^     __4a^  —  7,ax 
dx      ^/ ^a-x'^  —  2(7x3      \/4rt;2 — 2ax 

Placing  this  value  of  irx.  equal  to  zero  we  have 

4a 

X —  3 

In  order  to  determine  whether  this  value  of  x  corresponds 
to  a  maximum  or  minimum  of  the  function,  it  will  be  neces- 
sary to  find  the  sign  of  the  second  differential  coefficient. 

Before  doing  this  we  will  examine  a  method  by  which  the 
operation  may  be  somewhat  abridged. 

We  have  already  seen  that  when  the  value  of  a  function 
is  reduced  to  zero  by  giving  a  particular  value  to  the  varia- 
ble, it  does  notfollmv  that  its  differential  will  also  be  reduced 
to  zero  by  the  same  value  of  the  variable  (Art.  31),  for  the 
function  in  passing  through  zero  may,  and  probably  will,  be 
passing  from  negative  to  positive,  or  vice  versa,  and,  there- 
fore, may  have  a  differential  or  rate  of  change  at  that  point, 
the  same  as  at  any  other.  Thus  the  latitude  of  a  vessel  on 
the  equator  is  zero,  but  it  may  be  changing  as  rapidly  there 
as  anywhere  else. 


Il8  DIFFERENTIAL    CALCULUS. 

Now  if  we  wish  to  obtain  the  second  differential  coefficient 
for  a  particular  value  of  the  variable,  we  may  take  advantage 
of  this  circumstance.  Suppose  we  find  the  first  differential 
coefficient  to  be  the  product  of  two  or  more  factors;  either 
of  these  being  reduced  to  zero  will  reduce  the  coefficient  to 
zero.  In  this  case  we  may  obtain  the  value  of  the  second 
differential  coefficient  for  the  corresponding  value  of  the 
variable,  without  differentiating  the  entire  coefficient.  For 
suppose  we  have 

di-yy  y 

in  which  J/  .  y'  and  j"  are  functions  of  x;  this  product  will 
be  reduced  to  zero  by  any  value  of  x  that  will  reduce  either 
factor  to  zero.  Suppose  that  for  x=a  we  havej/=^y  if  we 
differentiate  the  function  we  have 

"^{f.)  J-'  uj{yyy")^y'/  dy     y y"  dy'    y  y'  dy" 
dx        dx^  dx  dx  dx      '       dx 

and  since  x^^a  reduces  j^  to  zero,  the  two  last  terms  of  this 
expression  become  equal  to  zero,  and  we  have 

d^  M _y' y" dy 
dx"  dx 

hence  to  obtain  the  value  of  -—^  for  that  value  of  x  that 

dx- 

reduces  one  factor  of  the  first  differential  coefficient  to  zero, 
we  have  only  to  multiply  its  differential  coefficient  by  those 
factors  which  do  not  become  zero,  and  then  substitute  the 
value  of  X.     If  for  example  we  have 

Tx=x{x^  —d^)—x{x-\-a){x^a^ 
we  may  reduce  it  to  zero  by  making 

x=^o,  x^a  or  x:= — a 

d^  u 
If  we  wish  the  value  of  -r-r  for  the  first  value  of  .t,  we  have 

dx^ 

a         ZC  an  o 

^x^ — (7''=  —  a^ 

uX    x  =  o 


MAXIMA    ANIJ    MINIMA. 


119 


If  for  the  second  value  we  have 


d^  u 


ax    x=a 
If  for  the  third  vahie  we  have 


^x{x-\-a)^=2a^ 


UX"   x=: 

Resuming  now  equation  (2) 
dS_4a^  —  xax 
dx 


=x{x—a)  =  2a 


^•jtt-.^ 


4'Z 


y^  4a'^ — 2ax      \/ ^a^  —  2ax 
which  becomes  zero  by  making  t^x^^u  or  x=-"-^^    we    shall 
have,  for  that  value  of  x^ 

d^S_  a  d{^a  —  ix)_        —^a 

dX^         \/ :^a^  —  2ax    '  dx  V  4ii2--~  2ax 

which  being  negative  shows  that  x=^  3-  corresponds  to  a  max- 
imum of  the  function. 

Hence,  i/  a  right  cone  be  tnsci'ibed  in  a  given  sphere  it  will 
have  the  greatest  possible  convex  surface  7vhen  the  axis  of  the 
£one  is  equal  to  tivo- thirds  of  the  diameter  of  the  sphere. 

Ex.  5.  A  point  0  (Fig.  11)  being  given  within  the  right 
angle  BAG,  through  which 
a  line  is  to  be  drawn  meeting 
the  axes  A  B  and  A  C,  it  is 
required  to  find  the  distance 
A  n  such  that  the  length  of 
the  line  between  the  points 
of  its  intersection  with  the 
axes  shall  be  a  minimum. 

Let  Km^a,  Oin^-b  and  inn^^x.      Then  the   right  angled 


triangles  Omn  and /A«  give  the  proportion  vin  :  O/n 
kp  or 

X  :  b  :  :  a-\-x  :  Ap 
whence 

X  x'^ 


An 


I20  DIFFERENTIAL    CALCULUS. 

But 


whence 
whence 


——2    ,  2       2 

A/    -f-  An   =  pn 


2       A2  (/z-^-Jc'l^ 


/;z=- V'^^  +-^2 

X 


which  is  the  function  of  which  we  are  to  find  the  minimum 

value. 

Representing  this  function  by  u^  and  considering  it  as  the 

product  of  two  factors,  we  have 

a-\-x     

u  — v'<^2  J^x^ 

X 

du      a-\-x  X         .     . — a 

dx         x        y/ tf^ +x^  x^ 

Reducing  to  a  common  denominator  we  have 

du      {a-\-x)x^  —a(ly^  -j-x^) _    x^—ab^ 
dx  x^^  b^^  +.r2  x^\'WTx^ 

Making  this  equal  to  zero  we  have 

To  find  if  this  is  a   minimum  we  differentiate   again,  but 

as  the  numerator  of  the   differential   coefficient  is   equal  to 

zero  for  this  value  of  x  (Ex.  4),  we  multiply  its  differential 

by 

I 


X'^  y^b""-  +X2 

which  gives 

d^u 3JC^         3 

dx^       x^\/  b'^  +x-^      'S/b'^+x'^ 
a  result  that  is   essentially  positive   whatever  may  be   the 

value  of  X.       Hence  x=^^y'ab^  corresponds   to    a  minimum 
length  of  the  Wwq  pn.     If  a  and  b  are  equal,  we  have 


MAXIMA    AND    MINIMA. 


121 


x^b  or  mn^=-Oin 
whence 

hp  —  \n 

Ex.  6.     To  find  the   maximum  rectangle  that  can  be  in- 
scribed in  a  given  parabola. 

Let  AC  B  (Fig.  12)  be 
the  parabola  of  which 
C  D  is  the  axis.  Let 
D  E  be  the  height  of  the 
inscribed  rectangle,  and 
let  CD=^  and  CE=jc, 
then     FE"  ==  2px     and 

FG=2\/2/'-f 

The  area  of  the  rec- 
tangle is 

FGxED  or  2^ ipx^a  —  x^ 
whicii  is  to  be  a  maximum. 

Dropping  the  constant  factor  2^  2/   (Ex.  2),  representing 
the  function  by  u  and  differentiating,  we  have 


u'=^ax 


%        ,  (ill       1      — 2" 
■x"  and  — ^=z-hax  "  ■ 

dx 


which  being  made  equal  to  zero  gives 


a 

A— 3 


hence  the  altitude  of  the  rectangle  is  equal  to  two-thirds  that  of 
the  parab  'la. 

To  show  that  this  is   a   maximum  we  differentiate  again, 
and  find 


./2  u  -^     -    -1 


dx"^ 


4  u>^^  4 


X 


which  is  negative  for  every  positive  value  of  .r,  and  therefore 

for  X  —  3- 

JSx.  7.     What  is  the  length  of  the   axis  of  the  maximum 
parabola  that  can  be  cut  from  a  given  right  cone? 


122 


DIFFERENTIAL    CALCULUS. 


Let  A  BC  (Fig.  1 3)  be  the  given  cone, 
and  FDH  the  parabola  cut  from  it. 
Let  DE  be  the  axis  of  the  parabola, 
and  AB  the  diameter  of  the  base  of 
the  cone.  Represent  AB  by  a,  AC  by 
/?,  and  BE  by  x.  Then  AE=a—x, 
FE=\/ax—x2.     Also 

AB  :  AC  : :  EB  :  ED  or  ^  :  ^ : :  ^  :  ED 
hence 

ED  =5,- 

But  the  area  of  the  parabola  is  equal  to 

xb 


fFHxDE  or|^ 


2\/ax — x2 

which  is,  therefore,  to  be  a  maximum.  Dropping  the  con- 
stant factor  3^  (Ex.  2),  representing  the  function  by  z^,  and 
differentiating  we  have 

(0 


dz/      T^ax'^ —  4.x 


u  —  ^  ax"^ — x^ 

3 


dx       2^ ax'^—x^        2^ ax^—x'^    '  ^^^"^        ^^  ^  ^^ ^ 

Placing  the  second  member  of  equation  (2)  equal  to  zero 
we  have  x=o  and  x  —  -^  ;  and  differentiating  (2)  for  this  last 
value  of  X  we  have  (Ex.  4) 


d'^  u 


{6ax—i2x^) 


(3) 


dx"^         2'\/ax^ — x^ 

Substituting  this  value  of  x  in  the   second  member  of  this 
equation  it  is  reduced  to 

d^  u  _ 

Hence  the  axis  of  the  maxi?nu7?z  parabola  is  three-fourths  of 
slant  height  of  the  cone. 

Ex.  8.  It  is  required  to  determine  the  proportion  of  a 
cylinder,  that  shall  have  a  given  capacity,  and  whose  entire 
surface  shall  be  a  minimum. 


MAXIMA    AND    MINIMA. 


123 


Let  a^  be  the  capacity  of  the  cylinder,  jr=4AB  (Fig.  14) 
the    radius  of    the    base,   and  y=AC 

be    the    height;    then    the    two   bases   CL - \t> 

taken  together  will  be  equal  to  2-.x-, 
and  the  convex  surface  to  2-xy,  so  that 
27:(x^-\-xy)  is  to  be  a  minimum.     Now 

r:x^y=a^,  whence  y= — —.      Substitu- 

ting  tkis  value  of  y,  representing  the   ^ 
function  by  z/,  and  differentiating  we 
have 


Fig.  14, 


U  =  2~X'  -\ 

X 
3 


dx 


2a' 


'.\-x— 


X' 


d  ^  u Ao^x 


(l) 
(2) 
(3) 


(!u 


Placing  the  value  of  ^i  equal  to  zero  we  have 

2-x'^^=^a^  ==~x"y 
whence 

y^2X 

or  f^e  height  of  the  cylinder  is  equal  to  the  diameter  of  the  base 

and 

a 


V^. 


This  value  of  x  corresponds  to  a  minimum  value  of  the 
function,  as  is   shown  by    substituting  it   in  equation  (3), 


which  gives 


d"  u 
dx" 


12: 


a  positive  quantity. 

Ex.  9.  To  divide  a  right  line  into  two  parts  such  that 
one  part  multiplied  by  the  cube  of  the  other  shall  be  a  max- 
imum, 

Ans.     The  part  cubed  is  three-fourths  of  the   given  line. 


124  DIFFERENTIAL    CALCULUS. 

Ex.  \o:  To  find  the  greatest  right  angled  triangle  that 
can  be  con-structed  on  a  given  line  as  a  hypothenuse. 

Alls.     The  triangle  must  be  isosceles. 

Ex.  II.  It  is  required  to  circumscribe  about  a  given  par- 
abola, a  minimum,  isosceles  triangle.  What  is  the  length  of 
its  axis."* 

Ans.     Four-thirds  the  axis  of  the  parabola. 

Ex.  12.  What  is  the  altitude  of  the  maximum  cylinder 
that  can  be  inscribed  in  a  paraboloid  ? 

Ans      Half  that  of  the  paraboloid. 

Ex.  13.  The  whole  surface  of  a  cylinder  being  given,, 
how  do  the  base  and  altitude  compare  with  each  other  when 
the  volume  is  a  maximum  .'*  A71S. 

Ex.  14.  Required  the  minimum  triangle  formed  by  the 
axis,  the  produced  ordinate  of  the  extreme  point,  and  the 
tangent  to  the  curve  of  a  parabola.  Ans. 


SECTION  V. 


APPLICATION  OF   THE  DIFFERENTIAL  CALCULUS  TO 
THE   THEORY  OF  CURVES. 


SIGNIFICATION    OF    THE    FIRST     DIFFERENTIAL     COEFFICIENT. 

(34)  In  order  to  form  such  a  conception  of  a  line  as  will 
be  adapted  to  the  methods  of  the  differential  calculus,  we 
must  consider  it  to  be  the  path  of  a  flowing  point. 

The  law  which  governs  the  movement  of  the  point  deter- 
mines the  nature  of  the  line,  as  to  form  and  position;  and 
this  law  is  expressed  in  the  Cartesian  system  l)y  the  equation 
which  shows  the  relation  between  the  coordinates  of  the 
generating  point  in  every  position  it  may  occupy  throughout 
its  movement. 

The  direction  in  which  the  point  is  moving  is  determined 
by  the  relative  rates  at  which  the  coordinates  are  changing 
their  values  at  the  same  moment.  If  the  rate  of  change 
should  be  constantly  the  same  in  each  of  the  coordinates, 
whether  negative  or  positive,  the  generating  point  would 
move  constantly  in  the  same  direction,  describing  a  straight 
line  ;  and  this  direction  would  be  determined  by  the  ratio  of 
these  rates,  which  in  this  case  would  be  measured  by  the 
simultaneous  increments  or  decrements  of  the  coordinates 
themselves. 

125 


126 


DIFFERENTIAL    CALCULUS. 


F,ig.  15. 


CO 


If,  for  instance,  the  coordinates  AB  and  BC  (Fig.  15)  have 
each  a  constant  rate  of  increase, 
the  ratio  of  the  increments  CO  and 
OC  will  be  constant,  and  the  gen- 
erating point  C  will  move  in  a 
straight  line,  whose  direction  will 
be  determined  by  the  re/afive  rates 
with  which  those  increments  are 
produced  ;  or  since  the  rates,  being 
uniform,  may  be  represented  by 
the   simultaneous   increments,  the 

direction  will  be  determined  by  the  ratio  ^  ^ 

But  if,  while  the  rate  of  change  in  one  of  the  coordinates 
is  constant,  that  of  the  other  should  constantly  vary,  the 
ratio  of  their  simultaneous  increments  would  be  constantly 
changing  and  the  point  would  describe  a  curve,  whose  char- 
acter would  be  determined  by  the  law  which  should  govern 
these  varying  rates  of  change;  and  this  law  would  be  expressed 
in  the  equation  of  the  curve.  But  if  the  varying  rate  of  change 
in  the  coordinate  should,  at  any  point  in  the  curve,  cease  to 
vary,  and  should  continue  afterwards  constantly  the  same 
as  at  that  point,  the  generating  point  would  cease  to  describe 
a  curve,  and  would  move  in  a  straight  line  in  the  direction 
to  which  it  was  then  tending;  and  this  direction  would  be 
determined  by  the  ratio  between  the  rates  of  change  in  the 
coordinates  as  they  existed  at  the  instant  they  both  became 
Muiform. 

JVow  the  tangent  to  any  curve  is  the  line  which  would  be  des- 
cribed by  the  generating  point  if  it  were  to  move  in  the  direction 
to  which  it  is  tending  on  its  arrival  at  the  point  of  tangcncy  ; 
just  as  a  stone,  when  it  leaves  a  sling,  describes  a  line  tan- 
gent to  the  curve  in  which  it  was  moving  at  the  instant. 
Hence  the  ratio  of  the  rates  of  change  in  the  coordinates  of 


THEORY    OF    CURVES. 


127 


Fig.  16. 


any  point  of  a  curve   determines   the   direction  of  the  line 
tangent  to  the  curve  at  that  point. 

Suppose,  for   example,  that  AB   (Fig.  16)   has    a  uniform 
rate  of  increase,  while  BC   has  a 
rate  of  increase  that  is  constantly 
diminishing;  the    point   C   would 
describe  a  curve. 

Now  let  us  suppose  that  the 
generating  point  on  arriving  at  the 
point  C  of  the  curve  should  con- 
tinue to  move  in  the  direction 
toward  which  it  was  then  tending, 
and  should  with  a  uniform  motion,  at  the  same  rate  as  it  had 
at  C,  describe  the  right  line  CD,  then  this  line  would  be  tan- 
gent to  the  curve  at  the  point  C.  From  D  draw  a  line  DB' 
parallel  to  the  ordinate,  and  meeting  the  axis  of  abscissas  at 
B';  and  from  C  draw  a  line  parallel  to  the  axis  of  abscissas 
meeting  DB'  in  O.  The  triangle  CDO  will  have  important 
properties  which  will  require  careful  investigation. 

From  whatever  point  in  the  line  CD  the  line  DB'  is  drawn, 
the  ratio  between  the  lines  CD,  CO  and  DO  will  be  the 
same,  and  hence  as  CD  is  described  at  a  uniform  rate,  equal 
to  that  with  which  the  generating  point  is  moving  at  the 
point  C,  the  lines  CO  and  DO  will  also  be  described  at  a 
uniform  rate  equal  to  that  with  which  AB  and  BC  were  in- 
creasing at  the  same  instant.  Hence  the  three  lines  CD, 
CO  and  OD  are  the  increments  that  would  take  place  in  the 
arc,  the  abscissa  and  the  ordinate  in  the  same  unit  of  time,  at 
their  several  rates  of  change  existing  when  the  generating 
point  of  the  curve  is  at  C ;  and,  therefore,  CO  and  OD  may 
be  taken  as  symbols  representing  the  rate  of  increase  of  the 
abscissa  and  ordinate  of  the  curve,  while  CD  will  represent 
the  rate  of  increase  of  the  curve  itself  at  the  point  C ;  and 
is  at  the  same  time  tangent  to  it  at  that  point. 


128  DIFFERENTIAL    CALCULUS, 

So  that  if  we  designate  the  length  of  the  curve  by  s,  and 
consider  CD  as  representing  ds,  we  shall  have  CO=^:r  and 
OD— ^/i^  for  the  point  C  of  the  curve. 

The  tangent  of  the  angle  which  the  tangent  line  CD  makes 

with  the  axis  of  abscissas  is  equal  to  ^-q,  and  hence  calling 

this  angle  v. 

civ ,  /    \ 

7^-tang.  V  (i) 

and  since  CD   =CO   H-OU"  we  have 

iIj-  ^^dx"-\-dy'^  (2) 

We  shall  have  frequent  occasion  to  use  these  two  equa- 
tions in  investigating  the  properties  of  curves. 

(35)  The  usual  method  of  obtaining  equation  (i)  is,  to  sup- 
pose an  actual  increment  BB'  (Fig.  16)  given  to  the  abscissa, 
and  to  find  the  corresponding  increment  CO  of  the  ordinate 
from  the  equation  of  the  curve.  The  ratio  of  these  two  in- 
crements will  not  give  the  tangent  of  the   angle  z',  which  is 

equal  to  ^-^,  but  will  approach  it  as  the  increments  decrease, 

and  when  they    become  infinitesimal  or  vanish,  there  is  no 

A-ce  1      .  DO-,   CO 

dinerence  between  ^-q  and  ^-q. 

This  manner  of  reasoning  we  have  already  discussed  in 
the  introduction  to  this  work,  and  its  defects  have  been 
shown.  We  commit  an  error  in  giving  an  actual  increment 
to  the  abscissa  and  ordinate,  for  their  rates  of  increase  are 
not  obtained  from  any  actual  increase  in  value  (except  where 
the  rate  is  uniform),  but  from  the  law  of  change  derived 
from  the  equation  of  the  curve  ;  and  the  suppositive  incre- 
ments which  we  give  are  not  real,  but  symbolical,  repre- 
senting what  they  would  be,  by  the  operation  of  the  laia  con- 
trolling them  at  that  instant,  and  are,  therefore,  a  symbolical 
expression  of  that  law. 

The  truth  is  that  CO  and  OD,  so  tar  from  being  infinitely 
small  may  have  any  value  whatever  assigned  to   them ;    so 


THEORY    OF    CURVES. 


129 


that  we  may  consider  them  as  %'ariables  whose  simultaneous 
values  always  correspond  to  some  point  in  the  tangent  line. 
In  fact,  if  we  differentiate  the  equation  of  a  curve,  and  con- 
sider X  and  J'  as  constants  for  the  point  of  tangency,  dx  and 
dy  may  be  considered  as  the  variable  coordinates  of  the  tan- 
gent line,  with  the  origin  at  the  i)oint  of  tangency,  and  the 
axes  parallel  to  the  primitive  axes.  Under  these  conditions 
the  differential  equation  of  the  curve  becomes  the  equation 
of  its  tangent  line.  This  can  easily  be  shown  by  a  few 
examples. 

Ex.  I.  The  equation  of  the  circle  with  the  origin  at  A 
(Fig.  17)  is 

y^^=-2'^x  —  x'^ 
whicn  being  differentiated  be- 
comes 

ydy = ( R  —  jc)(^/jc 
If  now  we  consider  x  and  y  as 
constants  for  the  point  P,  and 
i/A-(  =  PE)  and  ^/f(  =  ET)  as 
variables,  we  will  replace  the 
first  by  ;;/  and  ;/,  and  the  latter 
by  X  and  J,  and  we  have 

?;/y  =  (R  —  n)x 

which  is  the  equation  of  the  tangent  line  PT  with  the  origin 
at  P,  while  PE  and  TE  are  the  abscissa  and  ordinate  of  the 
line,  and  ///  and  n  are  coordinates  of  the  new  origin  referred 
to  the  primitive  one,  or  AB  and  BP. 

Suppose  now  we  transfer  the  origin  to  C,  the  center  of  the 
circle.  The  formulas  for  transferring  to  a  new  origin  in  a 
system  of  parallel  axes  are 

y^b-\-y   and  x^=^a-\-x' 
where  a  and  b  are  the   coordinates  of  the   new  origin.       In 
this  case  a  is  equal  to  BC=AC  — AI)  =  R  — ;^,  and  b  is  equal 
to  PB  =  —  ///,  and  hence  by  substitution, 


Fig.  17. 


(0 


130  DIFFERENTIAL    CALCULUS. 

m{  —  m-i-y)=(R-n)(R-n-^x')  (2) 

Calling  x"  and   y"  the  coordinates  of  the   point  of  tangency 
for  the  new  origin,  we  have 

x'^  =  —  R-\-n  or  'R.—n=^—x'' 
also 

y"  =  BF=m 
Substituting  these  values  in  equation  (2)  we  have 

y"{-/+y)  =  -x'\-x"Jr^') 

whence 

yy"+x'  x"=x"^-\-/^=R^ 
or  dropping  the  accents 

yy'-i-xx"^=R^ 
which  is.  the  equation  of  the  tangent  to  the  circle,  the  origin 
being  at  the   center  and   ^"  and  y  the   coordinates  of  the 
point  of  tangency. 

£x.  2.      If  we   differentiate  the   equation  of  the   ellipse 
referred   to   its   center  and  t^ 

axes,  we  have 


A^ydy-{-B^xdx=^o 


_.^ 


Making  ^     g  Q 

y(=B¥)  and  :r(=OB)  Fig.  18. 

(Fig.  18)  constant  and  dy{=TE)  and  ^/^(  =  PE)  variables, 
and  replacing_)^byy  and  x  by  x",dx  by  x  and  dy  hyy,  we  have 

AYy-{-B^x"x=o 

which  is  the  equation  of  the  tangent  line  to  the  ellipse,  with 
the  origin  at  P,  the  point  of  tangency,  and  the  axes  parallel 
to  the  primitive  ones;  x"  andj^"  representing  the  coordinates 
of  the  new  origin  referred  to  the  primitive  one,  and  x  andji/ 
the  variable  coordinates  ot  the  tangent  line  referred  to  the 
new  origin. 

If  we  transfer  the  origin  back  to  the  primitive  one  we  shall 
have 

x=^a-\-x'  and7=^+y 


THEORY    OF    CURVES.  I3I 

where  a  and  b  are  the  coordinates  of  the  new  origin  —  that 
is,  the  center  of  the  ellipse.  But^  =  — ;^''  and  l?^=—y"{{oTa 
is  essentially  positive  while  x"  is  essentiall}'  negative,  and  b 
is  essentially  negative  while  y"  is  essentially  positive),  and 
substituting  these  values  for  x  and_y  we  have 

AVy  +  B-jcV=Ay^+B2^"2=A2B2 

or  dropping  the  accents 

Ay^+BV'jc^A^B^ 

Ex.  3,     Differentiating  the   equation  of  the   parabola  we 

have 

ydy  ^=pdx 

Representing  x^y,  dx  and  dy  by  x\y\  x  and^y  respectively,, 
we  have  by  substitution 

y  y^=^px 
for  the  equation  of  the  tangent  line  to  the  parabola,  with  the 
origin  at  the  point  of  tangency.      If  we   transfer  it   to    the 
vertex  by  making  j;=(^+y  and  x=^a-\-x\  in  which  b^=^—y'^ 
and  a^—x",  we  have 

-y"^+y'y"=px-px" 
in  which  x"  and/"  are  the  coordinates  of  the  point  of  tan- 
gency for  the  origin  at  A.     Hence  _)^"^  =2/;c",  and  substitut- 
ing we  have 

,    II  \     I   II .    f      ,11 

—  2px  -\-y  y  —px  —px 

or,  dropping  the  accents, 

yy"  ^px  -\-px"  ^p{x  -}-  x"^ 
which  is  the  equation  of  the  tangent  to  the  parabola  at  the 
point  whose   coordinates  are  x"  and  y\  the  origin  being   at 
the  vertex. 

Ex.  4.     Lastly  we  will   take  the  equation  of  the  Hyper- 
bola referred  to  its  center  and  asymptotes 

A2  +  B2 

XV'— 

4 

from  which 

xdy-\-ydx'^o 


132  DIFFJiRENTlAL    CALCULUS. 

Replacing _)'  =  PB  (Fig.  19)  by  y,  ^=AB  by  x'\  dy  =  ET 
by  y  and  dx=^EF  by  x,  we  have 

y  x-\-x  y=-o 
in  which  x"  and  y"   are   the    coordinates  of  the   new  origin 

referred  to  the  primitive  one, 

and  X  and  y  are  the  variable 

coordinates   of    the    tangent 

line  TP  ;  the  origin   being  at 

P   and  the   coordinate  axes, 

PM  and   PN,  parallel  to  the 

asymptotes. 

To  transfer  the  origin  to  A 

the  center  of   the  hyperbola  Fig.  19. 

make  JF=<^+J^'^  and  x^=a-^xj   b  being   equal   to    —y"  and 

.a  =  —x";  and  substitute  these  values  for  x  and  y.       This 

gives 

y  {x  —X  )-\-x  yy  —y  )— o 

or,  dropping  the  accents 

/  A2-I-B3 

// 7      I,  II  \  II    \  II ^^       1^  -" 

y — y  = jr{^ — -^  )  oryx  -\-xy  — 

X  2 

which  is  the  equation  of  the   tangent  line  to  the  hyperbola 

referred  to  its  center  and  asymptotes. 

Thus  it  clearly  appears  that  differentials  are  not  infinitely 

small  quantities,  but  are  symbols  to  express  the  rates  or  laws 

of  variation^  which  are,  in  fact,  variable  functions  of  the 

GIVEN  VARIABLES. 

SIGN    OF    THE    FIRST    DIFFERENTIAL    COEFFICIENT 

(36)  If  X  and  J  represent  the  coordinates  of  any  curve, 
and  while  x  increases  uniformly  y  should  have  a  positive 
value  and,  also,  increase^  its  differential  will  be  positive  and 
the  curve  will  tend  to  leave  the  axis  of  abscissas  in  a  posi- 
tive direction  ;  but  if  y  should  be  decreasing  while  its  value 


THEORY    OF    CURVES. 


133 


is  positive,  its  differential  will  be  negative  and  the  curve  will 
approach  the  axis  of  abscissas  on  the  positive  side. 

Again  if  y  has  a  negative  value  and  increasing,  its  differen- 
tial will  be  negative,  and  the  curve  will  be  receding  from  the 
axis  of  abscissas  on  the  negative  side;  while  if  it  is  decreas- 
ing (being  still  negative)  its  differential  will  be  positive,  and 
the  curve  will  be  approaching  the  axis  of  abscissas  on  the 
negative  side  (see  definition  of  a  differential,  Art.  3). 

Hence  the  following  rule: 

When  the  ordinate  and  its  first  differential  have  the  same 
sign  the  curve  is  receding  fro77i  the  axis  of  abscissas,  and  when 
they  have  different  signs  the  curve  is  approaching  that  axis. 

Note. — The  differential  of  the  independent  variable  is  supposed  to  hz  constant  and 
positive,  and  hence  the  sign  of  the  differential  coefficient  is  the  same  as  that  of  the 
differential  itself  of  the  function. 

This  rule  may  be  illustrated  by  means  of  the  circle  (Fig. 
20)  whose  equation  (the  origin  bein^g  at  A)  is 

y'^  =2R^  —  .v^ 
from  which  we  obtain 

^v= dx 

y 

We  see  here  that  from  A  to  C,j  and  its  differential  have 
the  same  sign,  and  the  curve  recedes 
from  the  axis  of  abscissas.  The  same 
is  true  of  the  curve  from  A  to  D. 
But  from  C  to  B,  and  from  D  to  B, 
where  x  is  greater  than  R,  the  sign  of 
y  will  be  contrary  to  that  of  dy,  and 
the  curve  approaches  the  axis  of 
abscissas  on  both  sides. 

We  arrive  at  the  same  result  if  we 
consider  ^  as  representing  the  tangent  of  the  angle  which 
the  tangent  line  makes  with  the  axis  of  abscissas.  From  A 
to  C  and  from  A  to  D,  where  the  curve  leaves  the  axis  of 


134  DIFFERENTIAL   CALCULUS. 

abscissas,  the  sign  of  the  tangent  and  of  y  are  alike ;  while 
from  C  to  B  and  from  D  to  B  their  signs  are  contrary. 

fill  T?^  _  (Y* 

At  the  point  A  v^h.QXQ  y=o,  the  value  of  ^=—7;—  becomes 
infinite,  and  the  curve  departs  at  right  angles  from  the  axis 
of  abscissas.  While  at  the  points  C  and  D  the  value  ^ 
hecojiies  zero,  and  the  curve  neither  approaches  nor  recedes 
from  the  axis  of  abscissas.  And  this  corresponds  with  the 
value  of  the  tangent  of  the  angle  made  by  the  tangent  line 
with  the  axis  of  abscissas;  at  A  and  B,  1^-;^;=  00  and  the 
angle  is  a  right  angle  ;  at  C  and  D,  di=°  and  the  angle  is 
zero;  the  tangent  line  being  parallel  to  the  axis  of  abscissas. 


'^: 


SECTION   VI. 


DIFFERENTIALS   OF  TRANSCENDENTAL   FUNCTIONS. 

Proposition  I. 

(37)  To  find  the  differential  of  a  constant  quantity  raised 
to  a  power  having  a  variable  exponent. 

Let  the  constant  quantity  be  represented  by  a  and  the 
variable  exponent  by  v;  then  the  function  will  be 

if 
If  we  add  an  increment  to  v  which  we  will  call  in  we  shall 
have 

Differentiating  this  equation  we  have 

and  dividing  equation  (2)  by  equation  (i)  we  have 

^v  +  m  —  ^w  (3) 

This  equation  being  true,  irrespective  of  any  particular 
value  of  /;/,  it  will  be  true  for  any  value  we  may  assign  to  it; 
hence  the  differential  of  a  constant  quantity  raised  to  a 
power  denoted  by  a  variable  exponent,  divided  by  the  power 
itself  is  a  constant  quantity,  or 

daP 

But  we  have  seen  (Art.  6)  that  the  differential  of  the  vari- 


136  DIFFERENTIAL   CALCULUS. 

able  is  always  a  factor  in   the   differential  of  the  function. 
Hence  dv  will  be  a  factor  of  C.      Calling  the   other  factor 

k  we  have 

C^kdv 
whence 

da" 

-:^=-kdv 

or 

dd'=aVidv 

The  problem  now  is  to  find  the  value  of  k.       For   this  pur- 
pose we  expand  d'  by  Maclaurin's  theorem  and  have 

d'^A^Bv-^Cv^+nv^+Ev^-^-Qtc  (i) 

in  which 

dd"    ^    d^d"    ^        d^d' 
^^^^^=^'  ^=^'  ^==^77:;^'  '''■ 

when  V  is  made  equal  to  zero. 
But  we  have  found 

ddP 


hence 


or 


dv 

d[^)=dd'k=d'k^dv 
d^d" 


dv^ 


^'/^2 


and  similarly 


from  which  making  v=o  we  have 

^  =  1,  B=k,  C= — ,  /;= ,  £= ,  etc. 

'  '  2  '  2.3'         2.3.4' 

ISubstituting  these  values  in  equation  (i)  we  have 

aP  =  ^j^kvJ^ f- 1- 4-etc. 

'2        2.32  .3.4 

and  making  z^=-jt  this  becomes 


TRANSCENDENTAL  FUNCTIONS.  I  37 

a^  =i  +  i+^+2-73+o73-i+etc.=2.7i82  82  + 
If  we  represent  this  number  by  e  we  have 

a^  =e  or  a^=e^ 
If  e  is  made  the  l)ase  of  a  system  of  logarithms  /C'  would  be 
the  logarithm  of  a  to  that  base. 

This  was  done  by  Napier,  the  inventor  of  logarithms,  and 
the  system  having  that  base  is  called  the  Naperian  system. 
We  shall  indicate  the  logarithms  of  that  system  by  the  nota- 
tion /(?g.^  while  the  logarithms  of  other  systems  will  be 
noted  by  Zog.     We  have,  therefore, 

da)^^cf  log.  adv. 
that  is, 

The  differential  of  a  quantity  raised  to  a  poT.ver  denoted  by  a 
variable  exponent^  is  equal  to  the  power  multiplied  by  the  Nape- 
rian logarithm  of  the  constant  quantity  into  the  differential  of 
the  exponent. 

Proposition  II. 

(38)  To  find  the  differential  of  the  logarithm  of  a  vari- 
able quantity. 

Let  the  quantity  be  represented  by  r  and  its  logarithm  by 

v^  the  base  of  the  system  being  represented  by  a.      Then 

we  have 

r^=a^  and  dr=-a^'kdv 

whence 

dr  I        dr 

dv=^     ,,,    or  d  Lo^.  r^^-j  ■  — 
a  k  ^  k        r 

Representing  ^  by  M  we  have 

d  Lo^r_  r=M- 

in  which  Af  is  the  reciprocal  of  the  Naperian  logarithm  of 
the  base  a,  and  is  called  the  modulus  of  the  system  of  loga- 
rithms of  which  a  is  the  base.     Hence 


138  DIFFERENTIAL    CALCULUS. 

The  differential  of  the  logarithm  of  a  variable  quantity  is 
equal  to  the  modulus  of  the  system  to  whieh  the  losrarithm  be- 
longs^ into  the  differential  of  the  quantity  divided  by  the  quantity. 

In  the  Naperian  system  the  modulus  is,  of  course,  one. 
Hence  in  that  system 

d  log.  r=^-f 

from  which  we  learn  that  in  the  Naperian  system  the  rate  of 
increase  of  the  natural  number,  whatever  may  be  its  value, 
divided  by  the  rate  of  increase  of  its  logarithm,  is  always 
equal  to  the  number  itself. 

Note, — This  principle  was  used  by  Napier  himself  in  constructing  his  table  of 
logarithms,  and  explains  his  selection  of  his  peculiar  base.  Hence  he  is  one  of  the 
first  discoverers  of  the/r/«(:z]^/^of  the  differential  calculus,  although  he  never  applied 
it  otherwise  than  to  logarithms. 

(39)  If  we  call  e  the  base  of  the  Naperian  system  of 
logarithms,  a  the  base  of  any  other  system,  7;/  the  logarithm 
of  /  to  that  base,  n  the  Naperian  logarithm  of  /,  and  s  the 
Naperian  logarithm  of  a  we  shall  have 

p—a"'    p—e""     a~e' 

hence 

p—^sm—^n 

wherefore 

n  _i 
n=^s?n  or  7/^=7 — jn 

but  -^  is  the  modulus  of  the  system,  and  hence 

The  logarithm  of  a  number  in  any  systeni  is  equal  to  the  Na- 
perian logarithfii  of  that  number  ?nultiplied  by  the  ?nodulus  of 
the  system. 

This  property  is  not  peculiar  to  the  Naperian  system. 
The  logarithm  of  a  number  in  any  system  is  equal  to  the 
logarithm  of  the  same  number  in  the  common  system  mul- 
tiplied by  the  reciprocal  of  the  common  logarithm  of  the 
base  of  the  new  system. 

In  fact,  in  any  two  different  systems,  the  ratio  between  the 


TRANSCENDENTAL  FUNCTIONS. 


139 


logarithms  of  the  same  number  is  constant.  Thus  let  a  and 
b  be  the  bases  of  two  systems,  m  and  n  two  numbers,  and  x 
and  y  their  logarithms  in  the  first,  and  u  and  v  their  loga- 
rithms in  the  second  system  ;  then 

in^a^      ?i'—ay      ?/i^=b^      ?t^=b^ 
Avhence 

whence 


^  =b"    and  a'J  =W 

U  V 

a=^l?^   and  a=^by 


whence 


X       y 

or  the  ratio  between  the  logarithms  of  the  same  number  in 
two  different  systems  is  constant  and  equal  to  the  ratio  be- 
tween the  logarithms  of  any  other  number  taken  in  the  same 
systems.     Hence 

log.  a  :  coin.  Log.  a  :  :  log. 
whence 


10  :  com.Log. 


10^=1 


com.  Log.  a=i^^^Q=Al  log.  a 


as  we  have  seen. 


Proposition  III, 

(40)  To  find  the  differential  of  the  sine  of  an  arc. 

Let  APD  (Fig.  21)  be  a  circle  whose  center  is  at  O. 
POA  be  the  given  angle,  then  PB 
will  be  the  sine  of  the  arc  AP, 
and  also  an  ordinate  of  the  circle 
to  the  axes  OX  and  OY  ;  while  OB 
will  be  the  cosine  of  the  same 
I  angle,  and  also  the  abscissa  of  the 
point  P  of  the  curve,  and  AB  the 
versed  sine  of  the  angle. 

From  the  equation  of  the  circle 
with  the  origin  at  the  center  we 
obtain 


Let 


Fis.  21. 


I40  DIFFERENTIAL    CALCULUS. 

xdx^=- — -ydy 


and 


_y^dy'^ 


If  we  represent  the  arc  AP  by  s  we  have  (Art.  34) 

ds'^^^dx'^  -\-dy'^ 

whence 

^      y^dy^  iyX" -\-y") 

But 

whence 

ds^^= ^ — 

from  which  we  obtain 

X 

dy  ^^^^ds 

or 

cos,   s 
d  sm.  s  =  — ^^ — ds  ( I ) 

that'  is, 

The  differential  of  the  sine  of  an  arc  is  eqnal  to  the  cosine  of 
the  arc  into  the  differential  of  the  arc  divided  by  radius. 

Proposition    IV. 

(4 1 )  To  find  the  differential  of  the  cosine  of  an  arc. 
From  the  equation 

COS.  i'  =  VR^-sin.2  s 

we  have 

—  sin.  j.^/sin.  s 
dcos.  s^     , — ^       .     ^  — 
yR"  — sin."  s 

Substituting  the  value  of  <'/sin.  s  (Art.  40)  and  replacing  the 
denominator  by  cos.  s^  we  have 


TRANSCENDENTAL    FUNCTIONS.  14I 

—  sin.  j-cos.^  .^i"     —  sin.  J" 

t/cOS.  S  = 7S = iS ^^^  ( 2 ) 

R  COS.  s  R  ^   ' 

that  is, 

T/ie  differential  of  the  cosine  of  an  arc  is  equal  to  minus  the 
sine  of  the  arc  into  the  differential  of  the  arc^  divided  by  radius. 

Proposition  V. 

(42)  To  find  the  differential  of  the  tangent  of  an  arc. 

From  the  equation 

R  sin.  s 
tano;.  j'= 

°  COS.  s 

we  obtain  (Art.  14) 

R  COS.  s  .  d  ^\x\..  i"— R  sin.  s  .  d  cos.  s 

d  tang,  j-  = o ~ 

°  cos.*^  s 

Substituting  for  d  sin.  s  and   d  cos.  s  their  values  (Art.  40 

and  41)  we  have 

R  COS.-  5.^/j-  +  R  sin. 2  s.ds 

^tansf.  S  — —r. ;; ■ 

^  R  COS."  s 

or 

(cos. -J-  4-  sin.~i')^/j'         R^ 
^tangf.  6-~ s = r  ds  iv 

°  COS.-  S  COS-'i"  ^"^^ 

that  is 

The  differential  of  the  tangent  of  an  arc  is  equal  to  the  square 
of  the  raditis  into  the  differential  of  the  arc  divided  by  the 
square  of  the  cosine. 

Proposition   VI. 

(43)  To  find  the  differential  of  the  cotangent  of  an  arc. 

From  the  equation 

R- 

cot.  i"=T 

tang,  .f 
we  obtain 

—  R'^/tang.  s 

d  cot.  i'= — 7 ^ 

tang.~  s 


142  DIFFERENTIAL    CALCULUSc 

Substituting  for  i/tsnig.s  its  value  from  equation  (3)  we 
have 

<'/cot.  ^•= 2 — Z :r-  =  — — :7~  ^s  (4) 

COS.    i- tang.  •"  j-     sm."  JT  ^^ 

that  is 

T/ie  differential  of  the  cotangent  of  an  arc  is  negative^  and 
eqnal  to  the  differential  of  the  arc  multiplied  by  the  square  of  the 
radius^  and  divided  by  the  sqitare  of  the  sine. 

Proposition   VII. 

(44)  To  find  the  differential  of  the  secant  of  an  arc. 
From  the  equation 

R2 

sec.  i'= 

COS.  s 

we  have 

—  R^  ^/cos.  s 
d  sec.  j'= ^ 

COS.''  s 

Substituting  the  value  of  ^/ cos.  s  from  equation  (2)  we  have 

R  sin.  s 

d  SQC.  s^= 5 — ds  i^) 

COS.''  s  ^^^ 

that  is 

The  differential  of  the  secant  of  an  arc  is  equal  to  the  differ- 
ential of  the  arc  multiplied  by  the  radius  into  the  sine^  divided  by 
the  square  of  the  cosine. 

Proposition  VIII. 

(45)  To  find  the  differential  of  the  cosecant  of  an  arc. 
From  the  equation 

R3 

cosec.  s^=^—- 

sm.  s 

we  obtain 

—  R^  d  sin.  s 

d  cosec.  s^=- ■■ — s — 

sni.  "s 


TRANSCENDENTAL  FUNCTIONS.  143 

Substituting  the  value  of  ^sin.  s  from  equation  (i)  we  have 

—  R  COS.  s 

d  cosQC.  s— -■ — 5 ds  (6) 

sin.'^i"  ^   ^ 

that  is 

The- differential  of  the  cosecant  of  an  arc  is  equal  to  minus  the 

di'^crential  of  the  arc  multiplied  by  radius  into  the  cosine^  divided 

by  the  square  of  the  sine. 

Proposition  IX. 

(46)  To  find  the  differential  of  the  versed  sine  of  an  arc. 
From  the  equation 

ver.  sin.  j-  =  R — cos.  s 
we  have 

^  ver.  sin.  s^ — d  cos.  s 

Substituting  for  d  cos.  s  its  value  from  equation  (2)  we  have 

sin.  JT 
^ver.  sm.  j-= — .rz — ds  (7) 

that  is, 

The  differential  of  the  versed  sine  of  an  arc  is  equal  to  the 
differential  of  the  arc  multiplied  by  the  sine  and  divided  by  radius. 

(47)  In  these  equations  the  arc  is  supposed  to  be  thj 
independent  variable  ;  and  the  generating  point  is  supposed 
to  flow  around  the  circumference  at  a  uniform  rate. 

The  differential  of  the  arc  may  easily  be  found,  consider- 
ing it  as  a  dependent  variable  and  either  the  sine,  cosine, 
tangent,  etc.,  as  the  independent  one  varjang  uniformly. 

If  we  take  the  sine  as  the  independent  variable  we  have 
from  equation  (i). 

R  R 

ds= d  sin.  s^  ^d  gin.  s  (8) 

COS.  s  VR-  -sin.-jT 

If  we  take  the  cosine  we  have  from  (2) 

R  R  ,  , 

ds^  —  — d  COS.  s=^ —    ,  dcos.s         (9) 

sm.   s  VR~-cos.~j 


144  DIFFERENTIAL    CALCULUS. 

If  we  take  the  tangent  we  have  from  (3) 

cos.^  J'                       R^ 
ds^^ — T^ — d  tansf.  s^ d  tang,  s 

but 

sec."i-=R^  +tang.^i- 

hence 

R^  d  tang.i- 

R'^+tang.'-j-  ' 

Lastly,  if  we  take  the  versed  sine  we  have  from  equation  (7) 

R 

ds^=^''- ./  ver.  sin.  s 

sin.    ^' 

but  since 

sm.  S  =  /\/(2R  —  ver.  sin.  s)  ver.  sin.  s 
we  have 

R  d  ver.  sin.  s 

ds=    ,  .      ,  .    ~  (11) 

V  (2R— ver.  sm.  s)  ver.  sin.  s 

If,  in  equations  (8),  (9),  (10)  and  (it),  we  represent  sin.  s 
by  u,  cos.  s  by  x,  tang,  s  by  7,  and  ver.  sin.  s  by  s,  and  con- 
sider R  =  i,  we  shall  have 

du  ,     . 

ds^—^=^  (12) 

V  I— //^ 

^^=7^  (14) 

^■y=w r,         ■  (15) 

V  25  — S- 

From  these  equations  we  can  find  the  rate  of  change  in 
the  arc  when  we  know  that  of  either  of  the  four  trigonomet- 
rical lines. 


SIGNIFICATION    OF    THE     DIFFERENTIAL     EQUATIONS    OF    THE 
TRIGONOMETRICAL    LINES. 

(48)   Describe  the  circle  ADBE  (Fig.  22)  from  the  center 
O.     Draw  the  diameter   AB   and  the  tangent   TT'   at   its 


TRANSCENDENTAL    FUNCTIONS. 


145 


extremity.  Let  sc^  s'c\  s"c" ,  and  s"'c"'  be  the  sines  of  the  arcs 
As,  AD^',  ADB/'  and  ADBE/",  and  OT  andOT'  (negative 
for  ADs'  and  ADBj-")  be  the  secants  of  the  same  arcs ;  then 
AT  will  be  the  tangent  of  the  arcs  Ks  and  ADBjt",  and  AT' 
will  be  the  tangent  of  the  arcs  AD/  and  ADBE^"';  oc^  0^, 
oc"  and  oc"  will  be  the  cosines  of  the  same  arcs,  and  A*;,  A/, 
Kc"  and  k.c"  will  be  their  versed  sines. 

Suppose  now  the  generating  point  of  the  circle  to  move 
from  A  around  through  D,  B  and  E  back  to  A  with  a  uni- 
form rate  of  motion,  then 

From  equation  (i)  we  find  that  at  the  beginning  where 
cos.  =  R  the  rate  of  increase  of  the  sine  is  the  same  as  that 
of  the  arc  and  is  positive.  As  the  sine  increases  the  rate 
immediately  begins  to  decrease, 
being  always  in  proportion  to  the 
cosine,  until  the  generating  point 
arrives  at  D  and  the  cosine  be- 
comes zero.  The  sine  then  ceases  g 
to  increase,  being  at  a  maximum, 
and  its  rate  of  increase  is  zero. 

In  the  second  quadrant,  the  sine 
although  still  positive,  decreases, 
and  hence  its  rate  of  change  is 
negative  as  shown  by  the  cos.  s^ 
which  is  negative  in  that  quadrant, 
at  the  point  B  the  rate  of  decrease  has  become  equal  to  the 
rate  of  increase  of  the  arc,  although  the  value  of  the  sine 
itself  has  become  zero,  and  hence  at  that  point  ^/sin.  s-=^—ds. 
In  the  third  quadrant  the  sine  is  negative  and  increasing, 
hence  its  rate  is 'also  negative,  as  is  shown  by  the  cosine 
which  is  negative.  At  the  point  E  the  negative  increase 
ceases  and  the  rate  becomes  zero  as  does  the  value  of  the 
cosine.  In  the  fourth  tpiadrant,  while  the  sine  is  negative, 
it  is   diminishing,   and  hence  its  rate  of  change  is    positive 


Fig.  22. 


146  DIFFERENTIAL     CALCULUS. 

which  is  also  indicated  by  the  cosine  which  is  positive  for 
that  quadrant. 

From  equation  (2)  we  learn  that  in  the  first  and  second 
quadrants  where  the  sine  is  positive,  the  rate  of  change  in 
the  cosine  is  negative,  and  in  the  third  and  fourth  quadrants 
where  the  sine  is  negative  the  rate  of  change  in  the  cosine  is 
posi  ive.  We  learn  the  same  thing  from  the  figure,  for  from 
A  to  B  the  cosine  decreases,  being  positive,  or  increases, 
being  negative ;  while  from  B  to  A,  for  the  third  and  fourth 
quadrants,  it  decreases,  being  negative,  or  increases,  being 
positive ;  the  rate  of  change  being  at  all  times  in  direct  pro- 
portion to  the  value  of  the  sine. 

From  equation  (3)  we  learn  that  the  rate  of  change  in  the 
tangent  is  at  all  times  positive,  and  we  learn  the  same  thing 
from  the  figure,  for  as  the  arc  increases  from  any  point  what- 
ever, the  extremity  of  the  secant  which  limits  the  tangent 
will  move  upward  in  a  positive  direction,  and  the  tangent 
will  increase  in  the  first  and  third  angles  from  A  to  positive 
infinity,  and  decrease  in  the  second  and  fourth  from  nega- 
tive infinity  to  A ;  so  that  it  will  always  increase  positively 
or  decrease  negatively,  and  hence  its  rate  of  change  is  always 
positive  (Art.  3).  At  A  and  B  the  rate  is  the  same  as  that 
of  the  arc,  and  of  the  sine,  being  equal  to  ds^  while  at  D  and 
E  it  is  infinite. 

Similarly  we  learn  from  equation  (4)  that  the  rate  of 
change  in  the  cotangent  is  always  negative,  as  it  either 
decreases  being  positive,  or  increases  being  negative,  for  all 
points  of  the  circle. 

From  equation  (5)  we  learn  that  the  rate  of  change  in  the 
secant  has  the  same  sign  as  the  sine,  and  hence  is  positive 
in  the  first  and  second  quadrants,  and  negative  in  the  third 
'and  fourth.  By  inspecting  the  figure  we  see  that  in  the  first 
quadrant  the  positive  secant  OT  increases ;  in  the  second, 
the  negative  secant  OT'  decreases ;  in  the  third,  the  nega- 


TRANSCENDENTAL  FUNCTIONS.  147 

tive  secant  O  V  increases  ;  and  in  the  fourth,  the  positive 
secant  OT'  decreases.  At  the  point  A  the  rate  of  increase 
of  the  secant  is  zero  for  sin.  s=o,  while  the  secant  itself 
equals  R  (a  minimum).  We  see  also  that  the  rate  of  the 
secant  is  equal  to  that  of  the  tangent  multiplied  by  the  sine ; 
and  since  the  sine  is  (except  at  two  points)  always  less 
than  I,  the  tangent  increases  faster  than  the  secant  until  the 
arc  equals  90°,  when  the  sine  is  i  and  the  rates  become 
equal  being  infinite. 

Equation  (6)  will  give  a  similar  result  for  the  cosecant  in 
connection  with  the  cosine  and  cotangent. 

We  learn  from  equation  (7)  that  the  rate  of  increase  of 
the  versed  sine  corresponds  at  all  times  to  the  value  of  the 
sine,  and  is  therefore  positive  in  the  first  and  second  quad- 
rants, and  negative  in  the  third  and  fourth.  The  figure 
shows  tha4;  the  versed  sine  increases  positively  from  A  to  B, 
the  rate  of  increase  being  an  increasing  one  (corresponding 
to  the  sine)  in  the  first  quadrant,  and  then  a  decreasing  one 
in  the  second,  but  still  positive.  At  B  it  ceases  to  increase 
and  begins  to  decrease,  first  at  an  increasing  rate  in  the 
third  quadrant,  and  then  at  a  decreasing  rate  in  the  fourth, 
until  at  A  the  versed  sine  and  the  rate  both  become  zero. 
Hence  in  the  two  last  quadrants  the  rate  is  negative  corres- 
ponding to  the  sine,  while  the  versed  sine  is  always  positive. 
At  A  and  B  the  rate  of  change  is  nothing,  as  it  should  be, 
since  at  those  points  the  generating  point  of  the  circle  tends 
to  move  in  a  direction  perpendicular  to  the  line  on  which  the 
versed  sine  is  laid  off,  and,  therefore,  does  not  tend  to  alter 
its  value;  while  at  D  and  E,  the  generating  point  moves 
parallel  to  the  line  of  the  versed  sine,  and,  therefore,  at  those 
points  they  should  have  the  same  rate,  and  this  we  find  to 
be  the  case,  for  at   D 

sin.  s 
ii  VQT.  sin.  s=^ — -^ — /z^f—'A 


148  DIFFERENTIAL    CALCULUS. 

and  at  E 

sin.  J- 
^  ver.  sm.  s^ — ^5 — ds=^ — ^/s 

because  sin.  s  at  that  point  is  negative. 

VALUES    OF    TRIGONOMETRICAL    LINES. 

(49)  We  are  enabled  by  Maclaurin's  theorem  to  develop 
the  sine  and  cosine  of  an  arc  in  terms  of  the  arc  itself;  for 
let  s  be  the  arc  and  u  its  sine,  we  shall  have  (Art.  24) 

u=A-i-Bs-j-Cs^  +  Ds^ -{-  etc. 
and  (Art.  40,  41)  making  R  =  i  we  have 

i/u  d'^  u  .  d^  u 

—7  =cos.  s,      ,  o  =  — sin.  s,     J  o  =— COS.  s,  etc. 
ds  ds"  '    ds-^  ' 

making  ^  =  0  we  have 

fdu  \  fd^it  \  d^u 

(^^)=°'  Vdi)^'^  l':^J=°'  ~d^=-^^  ^^^- 

whence 

2/=sin.  s=^s — + ■  — ,  etc. 

2.3     2.3.4.5 

If  we  represent  by  ?/  the  cosine  s  then 

du  .  d'^t/  d"^  i(       , 

-T"  =  — sm.  s.     J  o  =  — COS.  s,     J  .,  =sin.  s.  etc. 

ds  ^    ds^  ds-^  ' 

Making  ^-=0  we  have 

(du  \  d^  1/  d^  ti 

('')=•'  1^^"°'  1^=-'^  1^""°^  ^'''- 

whence 

s^  s^  s^ 

2/=cos.  s^i  — — + — 7+,etc. 

2       2.3.4     2.3.4.5.6 

These  series  are  very  converging,  and' for  small  arcs  will 
give  the  length  of  the  sine  and  cosine  quite  accurately. 
In  order  to  apply  these  formulas,  we  take  the  length  of  a 

quadrant,  which  is  — ,  the  radius  being  i  ;  and  this  divided 


TRANSCENDENTAL  FUNCTIONS.  149 

by  90  and  then  by  60,  will  give  the  length  of  one  minute  of 

arc,  from  which  we  can  obtain  the  length  of  any  number  of 

minutes  or  degrees.     Substituting  the  value  of  the  arc  thus. 

found  in  the  formulas,  we  obtain  the  length  of  the  natural 

sine  or  cosine. 

If  we  wish  these  values  for  any  other  radius  we  shall  have 

for  sin,  s^^ti 

du      COS.  s     d^  11  _     sin.   y     d  ^  zi _     cos.  s 
^^~R~'    T^"^"    R=^    '    ~dF~~    R^ 

whence 


etc. 


u  —  s'u\.s=s  — i7T+ Z — t5~5~"   etc. 

2  .  3.  R-i      2  .  3  .  4.5  .  R^ 

(50)  We  may  in  a  similar  way  develop  an  arc  in  terms  of 
its  sine  and  cosine. 

Let  s  be  an  arc  whose  sine  is  z/,  then  (Art.  47) 
ds  I         ^"  s_  _| 

du    ^  \  —  u^'  du"       ^  ^ 

d^  s  3.  ,  ^_1 

^^=(i-«-)^+3«Hi-«-)   ■' 

making  ?/=o 

etc.,  hence 

i-^arc  whose  sine  is  ?/=//  +  — —4- — ' +  etc. 

2.3     2.4.5 

If  we  make  2^  equal  to  the  sine  of  30°  =|-  we  have  for  the 

value  of  the  arc 

s  =  h-\- ' ^  + r+ ^^ ^+   etc. 

"      2.3.2'^      2.  4. 5.2*'      2.4.6.7.2* 

the  sum  of  which  is  0.52359  nearly;  and  multiplying  this  by 

6  we  have  the  length  of  the  arc  of  a  semi-circle,  thus 

i8o°  =  "  =  3.i4i54  nearly 

which  is  also   the   approximate    ratio  of  the  diameter  of  a 

circle  to  its  circumference. 


SECTION   VII. 


OF    TANGENT   AND    NORMAL  LINES     TO    ALGEBRAIC 

CUR  VES. 

(51)  We  have  seen  (Art.  34)  that  when  x  and  7  represent 

the  abscissa  and  ordinate  of  the  curve,  ^  will  represent  the 

tangent  of  the  angle  made  by  the  tangent  line  of  the  curve 
with  the  axis  of  abscissas.  Now  the  equation  of  a  line 
drawn  through  any  given  point  is 

y^y  ^a{x — x) 
in  which  y'  and  x  are  the  coordinates  of  the  given  point, 
and  a  the  tangent  of  the  angle  made  by  the  line  with  the 
axis  of  abscissas.  Hence  for  any  curve  in  which  x'  and  y 
are  the  coordinates  of  the  point  of  tangency,  the  equation  of 
the  tangent  line  through  that  point  will  be 

y-y  -'^'\^—^ ) 

The  value  of  -7-'  will,  of  course,  be  obtained  from  the 
equation  of  the  curve,  and  by  substituting  that  value  we 
obtain  the  equation  for  the  tangent  line  of  that  curve. 

EXAMPLES. 

Ex.  I.     From  the  equation  of  the  circle  we  have 

dy  _      X 
dx  y 


TANGENT    AND    NORMAL    LINES.  151 

and  hence 

r 

y-y  =-^C^'--^^) 

or 

yy  -]-xx  =R2 

becomes  the  equation  of  the  tangent  line  to  a  circle. 

Ex.  2.     In  the  case  of  the  parabola  we  have 

dy  _p 
dx     y 
whence 

y—y  —~'\^—^'^ ) 

or 

yy'  —p[x-\-X^ 

becomes  the  equation  of  the  line  tangent  to  a  parabola. 
Ex.  3.     The  equation  of  the  ellipse  gives 


iv  V>-x 


dx!         h-^y 
whence 

y-y  -—jj^'K^-^) 

or 

becomes  the  equation  of  the  line  tangent  to  the  ellipse. 

Ex.  4.     From  the  equation  of  the  hyperbola  referred  to 
its  center  and  asymptotes,  we  have 

dy  _      y 
dx  X 

whence 

y-y  --^'(^^'-^'^) 

or 

yx  -\-xy  —        ; 


152 


DIFFERENTIAL    CALCULUS. 


becomes  the  equation  of  the  line  tangent  to  the  hyperbola, 
referred  to  its  center  and  asymptotes  as  coordinate  axes  — 
as  in  Art.  35. 

Since  the  normal  line  is  perpendicular  to  the  tangent,  if 
a  represent  the  tangent  of  its  angle  with  the  axis  of  abscis- 
sas, then  it  will  be  equal  to  — -^  where  a  represents  the  tan- 
gent of  the  angle  of  inclination  of  the  tangent  line.     Hence 

,  dx 

a  — — -77 
dy 

and  substituting  this  value  in  the  equation 

y—y  — a  \x — x  ) 
we  have 

f  ctX   /  t\ 

for  the  general  equation  for  the  normal  line,  and  it  may  be 
found   for  any  particular  curve  by   obtaining  the  value  of 


(fa;' 


from  the  equation  of  the  curve,  and  making  the  sub- 


stitution as  in  the  case  of  a  tangent  line. 


Proposition  I. 


(52)  To  find  the  general  expression  for  the  length  of  the 
subtangent  to  any  curve. 

Let  AP  (Fig.  23)  be  any  curve  of  which  PT  is  the  tangent 
at  the   point  P,  TB  the  subtangent, 
PN  the  normal,  and  PB  the  ordinate ; 
then  from  the  triangle  TPB  we  have 

PB=:-TB 
whence 

PB 


tang.  PTB 


TB  = 


tang.  PTB 


but 


dy 


A      B  N 

Fig.  23. 


tang.  PTB  =-7^  and  PB=y 

"  dx  -^ 


TANGENT    AND   NORMAL    LINES.  155 

hence 

that  is, 

The  subtangent  to  any  curve  is  equal  to  the  ordiuate  into  the 
differential  of  the  abscissa  divided  by  the  differential  of  the 
ordinate. 

Proposition  II. 

(53)  To  find  the  general  expression  for  the  length  of  the 
tangent  to  a  curve. 

From  the  triangle  PTB  (Fig.  23)  we  have 


J-    J-         V    PB  "  -|-  TB 

whence 

that  is, 

The  length  of  the  tangent  to  ajiy  curve  is  equal  to  the  ordinate 
into  the  square  root  of  one  plus  the  square  of  the  differential 
coefficient  of  the  abscissa. 

Note. —  V>y  the  '"''  length  of  the  tattgent''^  is  meant  that  part  of  the  tangent  line 
between,  the  point  where  it  intersects  the  axis  of  abscissas  and  the  point  of  tangency  on 
the  curve. 

Proposition  III. 

(54)  To  find  the  length  of  the  subnormal  to  any  curve. 
Since  the  triangle  PEN  (Fig.  23)  is  similar  to  the  triangle 
PBT,  we  have  the  angle  BPN=BTP,  and  hence 

BN  =  PB.  tang.  BPN 
or 

dy 


BN=v   , 

-^  dx 


that  is, 


154 


DIFFERENTIAL     CALCULUS. 


The  subnormal  is  equal  to  the  ordinate  into  the  differential 
coefficient  of  the  ordinate. 

Proposition  IV. 

(55)  To  find  the  length  of  the  normal  to  any  curve. 
Since  p^-  (Fig.  "^Z)  is  equal  to  Fb^+bn^,  we  have 


PN 


^Sj  y^^y^^-=^y\/i-\ 


dy' 


dx^ 


that  is, 

The  length  of  the  normal  line  is  equal  to  the  ordinate  into  the 
square  root  of  one  plus  the  square  of  the  differential  coeffcient 
of  the  ordinate. 

Note. —  By  the  '"'' length  of  the  nori7iaI''^  \s  meant  that  part  of  it  which  lies  be- 
tween the  point  of  its  intersection  with  the  axis  of  abscissas  and  the  point  of  the  curve 
to  which  it  is  drawn. 

(56)  The  following  examples  will  show  the  application  of 
these  formulas  to  particular  cases. 

Ex.  I.     From  the  equation  of  the  circle  we  have 

dx  y 

dy  X 

hence  the  subtangent  (Fig.  24)  is 


TB^y 


dx 
~dy 


PB 


X 


BO 


a   result   that   we   also  obtain 
from  geometry. 


Ex.   2.     The  length  of  the    t 
tangent  to  the  circle  is 


T¥=y\/,  +^=y\/^  +^  =|V  ^2  +y2  = 


,/y 

We  have  also  by  geometry 
TP :  PO 


Ry 

X 


PB:BO 


TANGENT    AND    NORMAL    LINES. 


155 


whence 


TP- 


PO  .  PB     Ry 


BO  X 

Ex.  3.     The  normal  line  of  the  circle  is 

Ex.  4.     The  subnormal  of  the  circle  is 

dy  x 

B0^v-7~  =  — y~  =  — ^ 

■^  ax  -"  y 

Ex.  5.     From  the  equation  of  the  parabola  we  have 

dx       y 

dy  ~7 

hence  the  subtangsnt  (Fig.  25)  is 

dx     y^      2px 
TB=v— 7-=— -= — — -  =  2.r=2AB 
■^  dy      p         p 


a   result  which    we    have    also 
from  geometry. 

Ex.  6.     The  tangent  of  die 
parabola  is 


Fijr.  2  = 


/  dx'^  /         y^  /  y^  

We  have  just  seen  (Ex.  5)  that  TB  =  2AB,  hence 

o  o  __  o 

TB"  — 4  AB'  — 4^" 

whence 

TP  =  V  y-'  +4.V-  =Vtb'  +  tb  "^ 
as  is  evident  from  the  figure. 

Ex.  7.     The  subnormal  to  the  parabola  is 

dv  p 

•^  dx     -^  y    -^ 
as  we  find  from  geometry. 


156 


DIFFERENTIAL    CALCULUS. 


Ex.  8.     The  normal  to  the  parabola  is 

■-y\/ 


PN 


dv' 


P' 


dx'^ 


y\/i  +Y  =  Vy"  +r-  -  V  PB 


'  + 


BN 


which  is  evident  from  the  figure. 


Ex.  9.     From  the  equation  of  the  ellipse  we  have 


B' 


X 


dy 

dx  A  ^7 

and  the  subtangent  (Fig.  26)  is 

dx  A~y^ 

-^  dy  B'^x 

This  value  for  the  subtangent 
does  not  contain  B,  and  hence 
is  the  same  for  all  ellipses  hav- 
ing the  same  major  axis,  the 
abscissa  being  the  same.  Hence 
the  tangent  to   the  circle  at  P 


A' 


X 


0       B 

Fig.  26. 

will  intersect  the  axis  of  abscissas  at  T,  and 


BT 


PB       OP   -OB       A2- 


X' 


OB 


OB 


X 


as  we  have  already  found. 


SECTION    VIII. 


DIFFERENTIALS   OF  CURVES. 

(57)  We  have  seen  (Art.  34)  that  the  differential  of  a 
curve  is  equal  to  the  square  root  of  the  sum  of  the  squares 
of  the  differentials  of  the  ordinate  and  abscissa.  Hence  to 
find  the  differential  of  any  particular  curve,  we  must  find 
from  its  equation,  the  differential  of  one  of  the  coordinates 
in  terms  of  the  other.  The  formula  will  then  give  the  dif- 
ferential of  the  curve  in  terms  of  a  single  variable. 

EXAMPLES. 

Ex.  I.     From  the  equation  of  the  circle  we  have 

xJx  xdx 

^  V 


in 


hence, if  we  designate  the  arc  by  //,we  have 

which  is  the  differential  of  the  arc  of  a  circle  in  terms  of 
the  variable  abscissa, 

Ex.  2.     From  the  equation  of  the  parabola  we  have 

dx^yfy 

and  calling  the  length  of  the  arc  //  we  have 

/ /     ,       y~i/v'      dv     , — ; 7 

du^^ dx^  ^dy-~  =  \/ dy'^  +-— y-=-j-V/-^  +J'- 

157 


DIFFERENTIAL    CALCULUS. 


Ex.  3.     From  the  equation  of  the  ellipse  we  have 

hence 

^^x^        _       ^^x^dx^        _   B^x'^dx^ 

'a-        ^  ^ 

hence 


du  ^=\/ dx^  -\- dj/^  =^  -rdxSJ 


DIFFERENTIALS    OF    PLANE    SURFACES, 

(58)  Every   surface   may  be  considered  as  generated  by 
the  flowing  of  a  line. 

If  we  wish  to  obtain  the  rate  at  which  the  surface  is  gen- 
erated we  must,  if  possible,  consider  every  point  in  the  line 
to  be  moving  in  a  direction  perpendicular  to  the  line  itself, 
if  it  is  straight,  or  to  its  tangent  at  that  point  if  it  is  a  curve. 
For  the  only  method  of  estimating  the  rate  at  which  the 
surface  is  generated  is  by  means  of  the  length  of  the  gener- 
atino;  line  and  the  rate  with  which  it  moves.  Now  unless  the 
movement  is  made  in  a  direction  perpendicular  to  the  line, 
the  rate  of  its  motion  will  be  no  criterion  of  the  rate  with 
which  the  surface  is  generated.  Thus  the  line  AB  (Fig.  27) 
moving  in  a  direction  perpendicular  to  itself  will  generate 
the  rectangle  ABCD,  but  if  it 
move  at  the  same  rate  in  any 
other  direction  as  A^,  the  surface 
generated  in  the  same  time  will 
be  less  until  if  it  should  move 
in  its  own  direction  it  would 
generate  no  surface  whatever, 
the  simple  movement  of  the  line  may  be  properly 
an  element  in  estimating  the  rate  of  generation  of  the 
surface,  it   must  always  be    supposed  to    take   place   in    a 


DIFFERENTIALS    OF    CURVES. 


159 


direction  perpendicular  to  the  line  itself  at  every  point. 
Otherwise  we  must  include  in  our  estimate  of  the  rate,  the 
sine  of  the  an^le  made  by  the  line  with  the  direction  in 
which  it  moves,  which  in  most  cases  would  be  inconvenient, 
and,  in  many,  impracticable. 

(59)  A  plane  surface  may  be  generated  in  two  ways  by  a 
straight  line  —  by  moving  so  as  to  be  always  parallel  to 
itself,  or  by  revolving  about  a  fixed  point.  If  it  is  supposed 
to  be  generated  by  the  first  method,  and  the  boundary  line 
is  symmetrical  about  the  axis  of  abscissas,  the  ordinate  of 
the  line  is  taken  as  the  generatrix,  and  while  it  moves  par- 
allel to  itself  one  of  the  extremities  is  in  the  line,  and  the 
other  in  the  axis,  and  thus  half  the  surface  is  generated. 

Thus  if  we  consider  AB,  the  diameter  of  the  circle  ADD 
(Fig.  28)  as  the  axis  of  abscissas,  we  would  consider  the 
upper  half  of  the  circle  as  generated  by 
the  ordinate  DE  moving  parallel  to  it- 
self, one  extremity  being  always  in  the 
curve  and  the  other  in  the  axis  AB. 
And  similarly  with  a  surface  bounded 
by  any  other  line  that  is  symmetrical 
about  the  axis  of  abscissas. 

(60)  The  differential  or  rate  of  increase  of  any  surface 
at  the  moment  the  generating  line  has  arrived  at  any  given 
position,  such  as  BC  (Fig.  29),  will  be  represented  by  the 
increment  that  would  take  place  (Art.  2)  in  ^ 
a  unit  of  time  if  the  surface  should  in- 
crease uniformly,  after  the  generating  line 
should  leave  the  position  BC  at  the  same  ^ 
rate  as  that  with  which  it  arrived  there. 
Now,  in  order  that  the  increment  may  be  uniform,  the  gen- 
erating line  must  maintain  the  same  length  and  flow  at  an 
unvarying  rate.  Thus  let  AC  be  the  curve  and  CB  the  gen- 
erating line  of  the  surface  ACB  ;  and  let  B/;  represent  the 


Fig.  28. 


l6o  DIFFERENTIAL    CALCULUS, 

uniform  increment  of  AB  in  a  unit  of  time,  at  the  same  rate 
as  at  B  ;  then  the  rectangle  Cci?B  will  represent  what  luoiild 
be  the  uniform  increment  of  the  surface  during  the  same 
unit  of  time  at  the  rate  at  which  it  was  increasing  at  CB. 
And,  hence,  if  we  consider  the  increment  B^  as  the  sym- 
bol representing  the  rate  of  increase  of  AB,  the  rectangle 
Cf/^B  will  be  the  proper  symbol  to  represent  the  rate  of  in- 
crease or  differential  of  the  surface  ACB.  But  the  rectan- 
gle is  equal  to  BC  .  V>b  j  and  if  we  call  AB  x,  BC  will  be  y, 
and  B<^  the  differential  of  x.  Hence  C<f/^B,  or  the  differential 
of  the  surface  will  be 

ydx 
that  is 

The  differential  of  a  plane  surface  bounded  by  the  axis  of 

abscissas  aiid  a  curve ^  is  equal  to  the  ordinate  multiplied  by  the 

differential  of  the  abscissa. 

(61)  In  order  to  obtain  the  differential  of  any  particular 
plane  surface  we  must  know  the  equation  of  the  line  that 
bounds  it,  in  order  that  we  may  eliminate  x  or  v  from  the 
formula.  We  shall  then  have  the  differential  of  the  surface 
in  terms  of  a  single  variable. 

Example   i. 

(62)  To  find  the  differential  of  a  triangle. 

Let  ABC  (Fig.  30)  be  the  triangle,  referred  to  A  as  the 
origin  and  AB  and  AD  as  coordinate  axes.  C 

The  equation  of  the  line  AC  is 

y^ax 
hence  . 

ydx^=axdx 
which  is  the  differential  of  the  surface  of        '       Fig.  30. 
the  triangle ;  a  being  the  tangent  of  the  angle  made  by  the 
line  AC  with  the  axis  AB. 


DIFFERENTIALS    OF    CURVES.  l6l 

Example  2. 

(63)  To  find  the  differential  of  the  surface  of  a  semi- 
circle. 

If  we  take  the  equation  of  the  circle  with  the  origin  at 
the  extremity  of  the  diameter,  we  have 

whence 


yi/x  —dx\/  2  Ra-  —  a-  ^ 
which  is  the  differential  of  the  surface  of  a  semicircle,  the 
origin  being  at  the  extremity  of  the  diameter.     If  we  take 


the  origin  at  the  center  we  have 


ydx =c/jC'v/  2^  2  _  ^  i 

Example  3. 

(64)  To  find  the  differential  of  the  surface  of  a  semi- 
ellipse. 

If  the  ellipse  be  referred  to  its  center  and   axes,  we  have 
from  its  equation 

B    . 

hence 

B        .— 

ydx  =  T ''/vv  A "  —  x ^ 

which  is  the  differential  of  the  surface  of  the  semi-ellipse. 

Example  4. 

(65)  To  find  the  differential   of  the  surface  of  a  semi- 
parabola. 

From  the  equation  of  the   parabola  referred  to  its  vertex 
■and  axis  we  have 

y—W  2j)x 


l62  DIFFERENTIAL    CALCULUS. 

hence 

_     JL 

ydx  =dx^/  2px = V  2/  x^dx 
which  is  the  differential  of  the  surface  of  a  serai-parabola. 

DIFFERENTIALS    OF    SURFACES    OF    REVOLUTION. 

(66)  A  surface  of  revolution  is  one  which  may  be  gener- 
ated by  a  curve  revolving  about  a  line  in  the  same  plane. 
Every  point  in  the  revolving  curve  will  describe  a  circle 
whose  plane  is  perpendicular  to  the  axis  of  revolution  and 
whose  center  is  in  the  axis.  Any  plane  passed  through  the 
axis  will  cut  from  the  surface  a  curve  which  is  identical 
with  the  revolving  curve. 

Such  a  surface  may  also  be  supposed  to  be  generated  by 
the  circumference  of  a  circular  section,  made  by  a  plane 
passed  through  the  surface  perpendicular  to  the  axis,  mov- 
ing parallel  to  itself  with  its  center  in  the  axis  of  revolution 
and  its  radius  varying  in  such  a  manner,  that  its  circumfer- 
ence shall  always  intersect  the  meridian  section  or  directing 
curve. 

The  rate  of  increase,  or  differential,  will  be  determined,  as 
in  other  cases,  by  finding  the  surface  that  would  be  generated 
in  a  unit  of  time,  if  the  generating  circle  were  to  move  dur- 
ing that  time,  without  change  of  magnitude  at  a  uniform 
rate,  equal  to  that  with  which  it  arrived  at  the  point  of  dif- 
ferentiation. Such  a  surface  would  be  equal  to  the  circum- 
ference of  the  generating  circle  into  the  line  which  repre- 
sents its  rate  of  motion.  Now  the  center  of  the  generating 
circle  is  supposed  to  move  along  the  axis  at  a  uniform  rate, 
hence  its  circumference  will  move  along  the  directing  curve 
at  the  same  rate  as  the  generating  point  of  the  curve  ;  so 
that  the  line  which  represents  this  rate  will  be  the  same  as 
the  differential  of  the  curve. 

Moreover  the  suppositive  differential  surface  that  we  are 


DIFFERENTIALS    OF    CURVES.  1 63 

seeking  must  be  generated  at  a  uniform  rate,  and  hence  the 
diameter  of  the  generating  circle  must  not  change;  so  that 
the  surface  will  be  that  of  a  cylinder,  whose  base  is  the  cir- 
cumference of  the  generating  circle  at  the  point  of  differen- 
tiation, and  its  height,  the  line  which  represents  the  differ- 
ential of  the  directing  curve  at  the  same  point. 

If  now  we  take  the  axis  of  abscissas  as  the  axis  of  revo- 
lution, the  radius  of  the  generating  circle  will  be  an  ordinate 
of  the  directing  curve  and  the  differential  of  the  curve  will 
t'G  V dx^  -\-(iy'^  (Art.  34);  and  hence  calling  the  surface  of 
revolution  S,  we  have 

that  is 

The  differ e7itial  of  a  surface  of  revoluiio7i  is  equal  to  the  cir- 
cu?nfcrence  of  the  geiurating  circle  into  the  differential  of  the 
directifig  curve. 

To  apply  this  formula  we  obtain  from  the  equation  of  the 
directing  curve,  the  value  of  one  variable  in  terms  of  the 
other,  and  by  substitution  obtain  the  differential  in  terms  of 
a  single  independent  variable. 

Example  i. 

(67)  To  find  the  differential  of  the  surface  of  a  cone. 

In  this  case  the  revolving  line  is  straight,  and  not  a  curve, 
but    the    principles    of    the    rule    apply 
equally  well. 

Let  AC  (Fig.  31)  be  the  revolving  ele- 
ment of  the  cone,  and  AB   the  axis  of  * 
revolution   and  of  abscissas,   the  origin 
being  at  A.     Then  we  have  for  the  equa-  Fig.  31. 

tion  of  the  line  KQ^y^ax  and  dy=adx,  a  being  the  tangent  of 
the  angle  BAG. 

Substituting  these  values  in  the  formula  we  have 
^ S  =  2 -ax^^ a-dx-  -{-dx-  =  2 -axdxy  ,;~^-|_i 


164  differential   calculus. 

Example  2. 

(68)  To  find  the  diiferential  of  the  surface  of  a  sphere. 

From  the  equation  of  tlie  circle  we  have 

xdx                ^     x"dx^ 
dy^=-  — and  dy"  ^ ^ — ■ 

y  y  ^  y.. 

hence 


whence 

d'^'=2~'Kdx 

for  the  differential  of  the  surface  of  a  sphere. 

As  the  entire  expression  besides  dx  is  composed  of  con- 
stants, we  infer  that  the  surface  of  a  sphere  increases  at  the 
same  rate  as  the  axis. 

Example  3. 

(69)  To  find  the  differential  of  the  surface  of  a  parabo- 
loid of  revolution. 

From  the  equation  of  the  parabola  we  have 

ydy  ,      y~dy" 

dx=^ — 7~  and  dx"  — — 7^ — 

hence  •  

^/S=:2-jV,/^2  4-,/^2  =  2-7/^-^       /    dy^ 

Example  4. 

(70)  To  find  the  differential  of  the  surface  of  an  ellipsoid 
of  revolution. 

We  found  (Art.  57)  that  the  differential  of  the  elliptic 
curv-e  is 


-rdx\/ 


A^-(A^-B~).T- 
A~  — ^^^ 


henceif  we  substitute  this  expression  in  place  of  V^/>:i:~+^/k^ 


DIFFF.RENTIALS    OF    CURVES.  1 65 

in  the  formula,  and  for^  its  value  derived  from  the  equation 
of  the  ellipse,  we  liave 


t/S  =  2-\/^,  (A2-x2)  .  -^dxsj 


A~  '  '      A      'V  A--JC^ 

which  becomes  by  reduction 

2~^dx    . 


DIFFERENTIALS    OF    SOLIDS    OF    REVOLUTION. 

(71)  A  solid  of  revolution  is  one  which  is  described  or 
generated  by  a  surface,  bounded  by  a  line  and  the  axis 
about  which  it  revolves.  If  this  axis  be  that  of  abscissas, 
then  the  ordinates  of  the  bounding  line  will  describe  circles, 
of  which  they  will  be  the  radii  and  the  centers  will  be  in 
the  axis.  Any  one  of  these  circles  may  be  considered  as 
the  generatrix,  which  describes  the  solid  by  moving  parallel 
to  itself,  as  in  the  last  case.  But  it  is  now  the  surface  of  the 
circle  and  not  merely  its  circumference  that  generates  ;  and 
its  movement  is  measured  along  the  axis,  the  rate  being  the 
same  as  that  by  which  the  abscissa  of  the  directing  curve  is 
increasing. 

Now  the  rate  of  increase  of  a  solid  of  revolution  is  meas- 
ured by  a  suppositive  increment  that  kwuLI  be  described  \vi.  a 
unit  of  time,  by  the  generating  circle  moving  uniformly 
along  the  axis,  with  its  diameter  unchanged  at  the  same 
rate  as  that  with  which  the  abscissa  is  generated.  Hence 
such  a  solid  would  be  a  cylinder  whose  base  is  the  genera- 
ting circle,  and  whose  altitude  is  the  line  representing  the 
differential  of  the  abscissa.  But  the  area  of  the  generating 
circle  is  ~y'^ ^  and  the  altitude  of  the  cylinder  is  dx  ;  hence 
the  cylinder  representing  the  differential  of  a  solid  of  revo- 
lution, would  be  expressed  by  the  function, 

-y'-dx 


l66  DIFFERENTIAL    CALCULUS. 

hence, 

The  differential  of  a  solid  of  revolution  is  equal  to  the  gener- 
ating circle  multiplied  by  the  differential  of  the  abscissa  of  the 
bounding  line. 

Example  i. 

(72)  To  find  the  differential  of  the  volume  of  a  cone. 

If  we  take  the  vertex  of  the  cone  for  the  origin,  and  the 
axis  of  abscissas  for  its  axis,  the  equation  of  the  revolving 
line  will  be 

y^ax 

and  hence  calling  v  the  volume  of  the  cone  we  have 

dv^^T^y"  dx^-a"  x'^  dx 

in  which  (a;  is  the  tangent  of  the  angle  made  by  the  revolv- 
ing line  with  the  axis,  and  x  the  distance  from  the  vertex  to 
the  base  of  the  cone. 

Example  2. 

(73)  To  find  the  differential  of  the  volume  of  a  sphere. 
If  we  take  the  origin  at  the  extremity  of  the  diameter,  the 

equation  of  the  revolving  semi-circle  will  be 

y'^  =2R;f — x'^ 
in  which  R  is  the  radius  of  the  sphere,  and  x  any  portion  of 
the   axis  of  revolution  measured  from  its  extremity  at  the 
origin  until  it  equals  2R;  hence  the  formula  for  the  differ- 
ential becomes 

dv^=^'^y'^dx=-T.{2^x — x'^^dx 

Example  3. 

(74)  To  find  the  differential  of  the  volume  of  an  ellipsoid 
of  revolution. 

If  we  suppose  the  semi-ellipse  to  revolve  about  its  major 


DIFFERENTIALS    OF    CURVES  167 

axis,  it  will  generate  an  oblong  ellipsoid  of  revolution,  oth- 
erwise called  a  prolate  s[)heroid.  If  we  take  the  origin  at 
the  extremity  of  the  transverse  axis,  the  equation  of  the 
ellipse  is 

y^  —  tt(2A.v— j;-) 

and  hence   the  formula  for  the  differential   of  the  volume 

becomes 

B- 

^i'  '=  ~y"dx  =  "  — ^  ( 2  A.r — a:-  )^r 

in  which  A  is  the  semi-transverse  and  B  the  semi-conjugate 
axis  of  the  ellipse  which  generates  the  ellipsoid  of  revolu- 
tion. 

If  we  take  the  conjugate  axis  of  the  ellipse  for  the  axis 
of  revolution  and  its  extremity  for  the  origin,  we  have 

A- 

j2=— 2(2Bx-.r-) 

and 

A2 
//e^=~-j7^(2B.r — x'^^dx 

In  this  case  the  volume  is  an  oblate  ellipsoid,  or  other- 
wise, an  oblate  spheroid. 

Example  5. 

(75)  To  find  the  differential  of  the  volume  of  a  parabo- 
loid of  revolution. 

The  axis  of  the  parabola  being  the  axis  of  revolution,  and 
the  origin  at  the  vertex,  we  have 

dv = ~j^ "  ^/jf  =  2  ~pxdx 

in  which  /  is  the  semi-parameter  of  the  revolving  parabola 
thai  iienerates  the  volume. 


SECTION  IX. 


POLAR     CURVES 


Proposition  I. 


(76)  To  find  the  tangent  of  the  angle  which  the  tangent 
line  makes  with  the  radius  vector. 

Let  CC'  (Fig.  32)  be  any  curve  of  which  we  have  the 
polar  equation.  Let  P  be  the  pole  ;  PM=r  the  radius  vec- 
tor; P(^  =  R,  the  radius  of  the 
measuring  arc  bd ;  dVd^  the  vari- 
able angle =zv  and  OT,  the  tan- 
gent to  the  curve  at  the  point 
M.  Produce  the  radius  vector, 
PM  to  R,  draw  RO  perpendic- 
ular to  PR,  meeting  the  tangent 
in  O  ;  draw  ON  parallel  to  RM, 
and  MN  parallel  to  RO,  meet- 
ing each  other  in  N.  Join  PN, 
and  drawc?^  parallel  to  MN  meeting  PN  in  a.  Suppose  the 
radius  vector  to  revolve  around  the  point  P  in  the  direction 
from  d  towards  b. 

The  generating  point,  being  supposed  to  have  arrived  at 
the  point  M  of  the  curve  will  be  subject  to  two  distinct, 
although  mutually  dependent,  laws  or  tendencies.  One  of 
these  tendencies  arises  from  the  law  of  change  in  the  length 

168 


Fig.  32. 


POLAR    CURVES.  169 

of  the  radius  vector,  which  causes  the  generating  point  to 
move  outward  in  the  direction  of  its  length.  The  other  ten- 
dency arises  from  the  revolving  motion  of  the  radius  vector 
which  causes  every  point  in  it  (including,  of  course,  the 
generating  point)  to  move  in  a  direction  perpendicular  to 
itself.  Hence  in  this  case,  the  law  would  incline  the  gen- 
erating point  to  move  in  tlie  direction  MN.  If  then  we 
take  the  distance  MR  to  represent  the  uniform  outward 
movement  that  would  take  place ^  under  the  influence  of  the 
first  law  in  a  unit  of  time,  it  will  represent  the  rate  of 
change  in  the  radius  vector  arising  from  that  law,  and  is, 
therefore,  the  symbol  of  that  rate  ;  that  is 

MR=.//- 
If  we  take  MN  to  represent  what  would  be  the  uniform 
movement  under  the  second  law  in  the  same  length  of  time, 
it  will  represent  the  rate  with  which  it  tends  to  move  in  the 
direction  MN  arising  from  that  law.  Now  as  both  these 
laws  act  together  without  disturbing  each  other,  the  gener- 
ating point,  if  left  to  its  tendency  at  the  point  M  would  move 
in  such  a  direction  as  to  obey  both  laws  or  influences  at  the 
same  time ;  and  hence  at  the  end  of  the  same  unit  of  time 
would  be  found  at  O,  having  described  the  line  MO  ;  the 
departure  from  the  line  MN  being  ON  =  MR,  and  the  depar- 
ture from  the  line  MR  being  RO=MN.  But  the  generating 
point  of  a  curve,  if  left  to  its  tendency  at  any  time  would 
move  in  a  line  tangent  to  the  curve,  and  since  the  line  MO 
would  be  uniformly  described  in  a  unit  of  time,  it  represents 
the  rate  of  increase  of  the  curve,  and  is  also  tangent  to  it. 
Hence  if  we  call  the  length  of  the  curve  //  we  have 

M0=./// 

The  point  b  at  the  intersection  of  the  radius  vector  with 

the  arc  of  the  measuring  circle,  tends  to  move  in  the  direction 

ba,  and  if  left  to  that  tendency  would  describe  that  line  in 

the  same  time  that  the  generating  point  would  describe  the 


170  DIFFERENTIAL    CALCULUS. 

line  MN;  for  the  rate  of  movement  of  b  is  to  that  of  M  as 
Yb  is  to  PM,  or  as  ba  is  to  MN.  If  then  we  consider  (^  as  a 
point  in  the  arc  of  the  measuring  circle,  we  may  consider 
ba  as  representing  its  rate  of  increase,  that  is  the  rate  of  in- 
crease of  the  angle  bVd^  and  hence 

ab^di^ 
But  from  the  triangles  PMN  and  Vba  we  have 

Yb\  PM:  '.ab\  MN 
hence 

PM  .  ab     rdv 

Now  the  tangent  of  the  angle  PMT=MON  is  equal  to 

MN_      MN 
^NO  "-'^llR 

and  substituting  the  value  of  MN  just  found  and  of  MR, 

we  have 

rdv 
Tang.  PMT=-^ 
^  dr 

that  is 

The  tangent  of  the  angle  which  the  line  tangent  to  a  polar 
curve  makes  with  the  radius  vector  is  equal  to  the  radius  vector 
into  the  differential  of  the  7neasuring  angle  divided  by  that  of  the 
radius  vector. 

(77)  Since  MNO  (Fig.  32)  is  a  right  angled  triangle,  we 
have 


MO   =MN   +N0   =MN  +MR 
hence  by  substitution 


r^dv'^ 
du^ 


R 

whence 


or  making  R  =  i 


& 


du=-^V  r-'dv^  +  'Khlr^ 

du  =  ^  r-dv-^dr^ 


POLAR    CURVES. 


171 


that  is 

The  differential  of  the  arc  of  a  polar  curve  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  radius  vector  into 
the  differential  of  the  measuring  angle ^  and  of  the  differential 
of  t?ie  radius  vector. 

Proposition    II. 

(78)  To  find  the  subtangent  of  a  polar  curve. 

The  subtangent  of  a  polar  curve  is  the  projection  of  the 
tangent  on  a  line    drawn   through    the 
pole  perpendicular  to  the  radius  vector 
of  the  point  of  tangency. 

Hence  if  PT  (Fig.  ^-^  be  drawn  per- 
pendicular  to    PM,    meeting   in    T    the 
tangent  to   the   curve   at  the  point  M, 
then  PT  will  be  the  subtangent.     Since 
MP.tan^.  PMT 
^  R 

we  have  by  substitution 


Fig.  33- 


PT=^  .  -T- 


rdv 
~d?'' 


*dv 


Kdr 


that  is 

The  subtangent  of  a  polar  curve  is  equal  to  the  square  of  the 
radius  vector  into  the  differential  of  the  ??teasuring  arc  divided  by 
R  into  the  differential  of  the  radius  vector.     If  we  make  R=;' 


we  have  PT: 


:^  =tan2;ent  of  PMT. 
dr  ^ 


Proposition  III. 

(79)  To  find  the  value  of  the  tangent  to  a  polar  curve 
The  tangent  to  a  polar  curve  is  that  part  of  the  tangent 

line  which  lies  between  its  intersection  with  the  subtangent 

and  the  point  of  tangency. 

Hence  MT  (Fig.  -^-^  will  represent  the  tangent,  and  since 


172  DIFFERENTIAL    CALCULUS. 


we  have 


Mr   =PM   4-  PT 


MT   =r^-^ 


or 


or  making  R  =  i  •  . 

Proposition  IV. 

(80)  To  find  the  subnormal  to  a  polar  curve. 

The  subnormal  of  a  polar  curve  is  the  projection  of  the 
normal  line  on  a  line  drawn  through  the  pole  perpendicular 
to  the  radius  vector  for  that  point  of  the  curve  to  which  the 
normal  is  drawn. 

Hence  if  MB  (Fig.  t,'^)  be  a  normal  at  the  point  M,  BP 
will  be  the  subnormal. 

The  triangles  MBP  and  MTP  being  similar,  the  angles 
MBP  and  PMT  are  equal,  and  since 

tancr.  MBP 


PM  =:BP- 


we  have 


r=BP 


R 

rdv 
"Kdr 


BP-     , 

av 

that  is 

The  subnormal  of  a  polar  curve  is  equal  to  radius  into  the  dif- 
ferefitial  of  the  radius  vector,  divided  by  the  differential  of  the 
measuring  arc. 

(81)  The  normal  line  MB  (Fig.  t^t^  is  equal  to 


POLAR     CURVES.  173 

or 


(82)  While  the  point  at  the  extremity  of  the  radius  vec- 
tor describes  the  line  of  a  polar  curve,  the  radius  vector 
itself  generates  the  surface  bounded  by  the  curve. 

Now  the  point  M  of  the  line  PM  (Fig.  32)  tends,  by  virtue 

of  the  revolving  motion  of  the  radius  vector  about  the  pole, 

to  move  in  the    direction  MN,  perpendicular  to  P^I,  and 

every  other  point  in  the  line  PM  will  tend  to  move  in  a 

direction  parallel  to  MN,  and  at  a  rate  proportional  to  its 

distance  from  the  fixed    point  P.      Hence  if   the   point  M 

were  to   be    found    at  N,  the    line  PM  would  assume  the 

position  PN,  and  the  triangle   PMN  would  be  that  which 

would  be  generated  at  a  uniform  rate  by  the  radius  vector 

PxM  if  left  to  its  tendency  when  in  that  position,  so  that  the 

triangle  PMN  is  the  true  symbol  to  represent  the  rate  at 

which  the  surface  bounded  by  the  [)olar  curve  is  generated, 

or,  designating  the  surface  by  O,  we  have 

triangle  PMN=^/0. 
But 

PMXMN 

PMN= . 

2 

and    substituting   here  the  values  already  found  for  these 
terms  we  have 

2K 

hence 

The  differential  of  a  su)-face  boujided  by  a  polar  ciirve  is 
equal  to  the  square  of  the  radius  vector  into  the  differential  of 
the  tncasuring  arc  divided  by  twice  its  radius. 

SPIRALS. 

(83)  If  a  right  line  revolve  uniformly  in  the  same  plane 
about  one  of  its  points,  and  a  second  point  should,  at  the 


174 


DIFFERENTIAL    CALCULUS. 


same  time  approach  to,  or  recede  from  the  fixed  point,. 
according  to  some  prescribed  law,  it  would  generate  a  curve 
called  a  spiral. 

The  fixed  point  is  called  the  polc^  and  the  curve  generated 
during  one  revolution  of  the  line  is  called  a  spire.  There 
being  no  limit  to  the  number  of  revolutions  of  the  line,  the 
number  of  spires  is  infinite,  and  a  line,  drawn  through  the 
pole  will  intersect  the  curve  in  an  infinite  number  of  points. 

Hence  there  can  be  no  algebraic  relation  between  the 
ordinates  and  abscissas  of  the  curve,  and  its  conditions 
must  be  expressed  by  a  polar  equation  which  will  be  in  the 
form 

in  which  r  is  the  radius  vector  and  v  the  measuring  arc  of 
the  variable  angle. 

SPIRAL    OF    ARCHIMEDES 


(84)  This  spiral  is  one  in  which  the  radius  vector  is  con- 
stantly proportional  to  the  corresponding  arc  which  measures 
its  angular  movement.     Hence  its  equation  will  be 

7'^av  (i) 

The  curve  may  be  constructed  in  the  following  manner. 
Divide    the    circumference    of  ""~~~~-~-^J' 

the  measuring  circle  into  eight  ^  l^^^^^r----^  T 

equal  parts  by  the  radii  AB, 
AC,  AD,  AE,  etc.,  (Fig.  34)  ;  / 
also  the  radius  AB  into  the 
same  number  of  parts.  Then 
lay  off  from  the  center  one  of 
these  parts  on  AC,  two  on  AD, 
three  on  AE,  and  so  on,  there 
being  eight  on  AB,  nine  on 
AC,  ten  on  AD,  and  so  on. 
Through  the  points  thus  found  draw  the  curve  commencing 


POLAR    CURVES.  175 

at  the  pole.     The  radius  vector  will  be  to  the  corresponding 
measuring  arc  as  the  radius  of  the  measuring  circle  is  to  the 

circumference ;  or  a  will  be  equal  to  — :;. 

Note. —  In  this  construction  we  have  supposed  the  radius  of  the  measuring  circle  to 
be  equal  to  the  radius  vector  after  one  revolution.  Of  course  any  other  proportion 
might  be  taken,  but  as  the  magnitude  of  the  spiral  does  not  depend  on  that  of  the 
measuring  circle,  the  radius  of  the  latter  may  always  be  taken  equal  to  the  radius  vec- 
tor after  one  revolution. 

If  we  differentiate  equation  (i)  we  have 

dr'=adv  (2) 

In  a  polar  curve  (Art.  76)  the  tangent  of  the  angle  which 
the  tangent  line  makes  with  the  radius  vector  is  equal  to 

dv 
dr 
and  from  equation  (2)  we  have 

dv       T 
dr  ~~  a 

hence  the  tangent  of  the  angle  APT  is  equal  to  -^,  or  in  this 

case,  to 

2rLr 

This  tangent  will  after  one  revolution  be  equal  to 

2-R 
(85)  The  subtangent  of  a  polar  curve  (Art.  78)  is 

r'^dv 

R^ 
which  becomes  for  this  curve 


R^z       R 
or,  making  r=R,  we  have 

subtangent = 2-R 
equal  to  the  tangent  of  the  angle  made  by  the  tangent  line 
with  the  radius  vector ;  and  also  ee|ual  to  the  circumference 
of  the   circle   described  by   the   radius  vector  as  a  radius, 


176  DIFFERENTIAL    CALCULUS. 

when  the  point  of  tangency  is  at  the  circumference  of  the 
measuring  circle. 

If  we  make  z^=;/2-R,  that  is,  if  the  tangent  be  drawn  to 
the  curve  after  n  revolutions  of  the  radius  vector,  then 

r  ;' 

V         7/2 -R 

whence 

dv       I      772  ~R  r"dv 

-7-=—= and  „   ,   =772 TTr 

dr       a         r  Ka?' 

that  is 

After  n  revolutions  of  the  radius  vector,  the  subtangent 
is  equal  to  77  times  the  circumference  of  a  circle  described 
by  the  radius  vector  as  a  radius. 

For  the  subnormal  whose  value  is  (Art.  80) 

R^/r 
dv 
we  have 

dr  __    _r 

dv  V 

hence 

Rr 

subnormal  = — 

V 

If  the  normal  is  drawn  at  the  point  B  then 

?;=2-R 

and  we  have 

r 
subnormal  =  — ' 

2~ 

that  is 

The  subnormal  is  equal  to  the  radius  of  a  circle  of  which 
r=R  is  the  circumference. 

THE    HYPERBOLIC    SPIRAL. 

(86)  The  equation  of  the  Hyperbolic  Spiral  is 

rv^=-ab 


POLAR    CURVES. 


177 


Fig.  35. 


in  which  r  is  the  radius  vector,  v  the  measuring  arc,  a  the 
radius  of  the  measuring  circle  and  b  the  unit  of  the  meas- 
uring arc — ab  being,  of  course,  a  constant  quantity.  It  is 
called  a  Hyperbolic  spiral  because  its  equation  resembles  that 
of  a  hyperbola  referred  to  its  center  and  asymptotes. 

To  construct  this  curve  describe  a  circle  with  a  radius 
PA  (Fig.  35) 
equal  to  a. 
Lay  off  from 
A  an  arc 
AB=^as  the 
unit  of  the 
ineasurin  g 
arc  v^  and 
continue  this 

■division  around  the  circumference  of  the  measuring  circle. 
Also  lay  off  A^'=^^,  Ar=^(^,  and  so  on. 

Through  these  points  of  division  in  the  circumference 
draw  the  radii  PB,  PC,  PD,  PE,  and  so  on,  and  produce  the 
radii  Vs  and  P/".  On  these  radii  lay  off  Vc^^\a^  Vd=-\a^ 
P6'=J(2,  and  soon  ;  also  P0  =  2^,  PQ=4a'.  Draw  the  curve 
through  the  points  thus  found. 

The  radius  vector  multiplied  by  the  measuring  arc,  count- 
ing from  A,  will  always  be  equal  to  the  radius  of  the  meas- 
uring circle  into  the  unit  of  the  arc,  that  is 

rv^^ab 

We  see  from  the  equation  that  r  increases  as  v  diminishes, 
and  vice  versa.  If  v^^o  r  becomes  infinite,  and  hence  the 
radius  vector,  through  A,  will  never  reach  the  beginning  of 
the  curve.  If  r'=^o  then  ?'  will  be  infinite,  hence  the  curve 
will  never  reach  the  pole. 

If  we  take  any  point  O  in  the  spiral  and  join  OP,  then 
OP  will  be  equal  to  r,  and  the  arc  A5=z'.  Draw  OR  per- 
pendicular to  PA  and  we  have 


lyS  DIFFERENTIAL    CALCULUS. 

OP  .  sin.  As     r  sin.  v 


0R=- 


PA 

OR.  a 

r= — ; — — ■ 
sin.  V 

OR  .  av 


sin. 

V 

-au 

OR  = 

sin. 

V 

-h 

hence 

and  since 

we  have 

whence 


V 

As  sin.  V  is  always  less  than  27,  the  line  OR  will  always  be 
less  than  b^  but  may  be  made  to  approach  that  value  as  near 
as  we  please.  Hence  if  we  draw  a  line  MN  parallel  to  PA, 
at  a  distance  from  it  equal  to  ^=arc  AB,  it  will  be  an  asymp- 
tote of  the  curve. 
Since 

dv  _      V 
dr  r 

the  subtangent  (Art.  78) 

r^dv TV 

^dr  a 

and  since 

rv^=^ab 
we  have 

subtangent  =— ^  =  — arc  AB 

a  constant  quantity.     Thus  P;//  or  P;/=AB  or  PM 

Also  the   tangent  of  tlie  angle  made  by  the  tangent  line 

with  the  radius  vector  (Art.  76). 

rdv 

that  is, 

The  tangent  of  the  angle  which  the  tange?itcine  makes  with  the 
raditcs  vector  is  negative  and  equal  to  the  arc  ichich  measures  the 
angle  made  by  the  radius  vector  with  the  fixed  line  PA.     Hence 


POLAR    CURVES. 


179 


this  angle  is  obtuse  on  the  side  of  the  radius  vector  toward 
the  origin,  while  the  subtangent,  being  also  negative,  lies  on 
the  side  opposite  to  the  origin. 

THE    LOGARITHMIC    SPIRAL. 

(87)  The  equation  of  the  logarithmic  spiral  is 
z'  =  Log.  r 
in  which  v  represents  the  measuring  arc  and  r  the  radius  vector. 

The  equation 
may  also  be  put 
into  the  form         F 

a'°  =/' 
the  relation  be- 
tween V  and  r  be- 
ing such  that 
while  V  increases 
in  arithmetical 
progression  rwill 
increase  in  gco- 
vietrical  progres- 
sion. Hence  the 
curve  may  be  con- 
structed by  lay- 
ing off  on  the  Fig.  36. 
measuring  arc  the  equal  distances  A/^,  hc^  cd,  de^  and  so  on 
(Fig.  '^(i)^  and  through  the  points  of  division  drawing  the 
radii  P^,  P^,  P^/,  P^,  and  so  on,  producing  them  if  necessary. 
On  these  radii  lay  off  the  distances  PB,  PC,  PD,  PE,  and  so 
on  in  geometrical  progression,  so  that 
PB_PC_PD_PE 
PA~PB~PC~PD 
and  through  the  points  thus  found  draw  the  curve.  That 
part  of  the  curve  within  the  circle  will  be  found  by  laying 
off  on  the  radii  P^/,  P/,  Yo.  and  so  on,  distances  from  P  by 
the  s-ame  rule,  and  thus  points  of  the  curve  may  be  found. 


=  ;Fr7^=T:^  and  so  on 


l8o  DIFFERENTIAL    CALCULUS. 

If  we  make  the  radius  of  the  measuring  circle  equal  to  i, 

and  reckon  the  arc  v  from  the  line  PA,  then  the  curve  will 

pass  through  the  point  A,  for  when  v^o  we  have 

log.  /'=^=log.  I 

and   if  we  call  a  the  ratio  between  PA  and  PB,  we  shall 

have 

PB=^,  PC=^%  PD=«^  PE=^^,  etc. 

when  the  exponent  is  always  equal  to  the  number  of  divis- 
ions of  the  measuring  arc,  and  is  therefore  represented  by 
the  arc  itself  corresponding  to  the  radius  vector,  whence 
a'o  =;^  or  7;=  Log.  /-  to  the  base  a. 
If  we  differentiate  the  equation  of  this  curve  we  have 

dv—yi —  (2) 

whence  (Art.  76) 

rdv       rMdr 

tang.  PDT  =—1-  —^-^  =  M 
°  ar         rar 

that  is 

The  tangent  of  the  angle  made  by  the  tangent  line  with  the 
radius  vectoj'  is  constant  and  equal  to  the  modulus  of  the  system 
of  logarithms  to  which  the  curve  belongs^  If  the  system  is 
the  Naperian,  M  =  i  and  the  angle  PDT  is  equal  to  45°. 

The  formula  for  the  subtangent  of  a  polar  curve  (Art.  78) 

is 

r'^dv 

and  substituting  in  this  the  value  of  ^-^|  from  equation  (2)  we 
have  (R  being  i) 

subtan.  = =  rM 

;■ 

If  M=i  then  subtang.  =r. 

For  the  value  of  the  tangent  we  have  (Art.  79) 

tang.  =  Vr-  +r2M-  ='V  i  4-M- 

If  M  =  i   then 

tang=r\/  2 


POLAR    CURVES.  l8l 

For  the  subnormal  we  have  (Art.  80) 

dr       r 

subnormal  =-7~  =t> 
dv      M 

If  M  =  i  then 

subnormal  =r 

For  the  value  of  the  normal  (Art.  81)  we  have 


normal  =  Y /-2  -f  ^^^ry  i  +^^3 

If  M  =  i   then 

normal  ^r-y/ "2 

These  values  show  that  these  lines  are  all  in  direct  pro- 
portion to  the  radius  vector.  The  same  result  flows  from 
the  constancy  of  the  angle  made  by  the  radius  vector  with 
the  tangent  line.  For  all  the  triangles  formed  by  the  radius 
vector,  the  tangent,  and  the  subtangent  will  be  similar  to 
each  other,  at  whatever  point  of  the  curve  the  tangent  may 
be  drawn.  The  same  may  be  said  of  the  triangles  found  by 
radius  vector,  normal  and  subnormal.  Hence  these  lines 
will  always  be  in  proportion  to  the  radius  vector. 

To  construct  a  logarithmic  spiral  for  a  given  base,  des- 
cribe a  circle  with  a  radius  equal  to  a  unit  of  the  radius 
vector,  PA,  and  lay  off  the  arc  Kb  equal  to  a  unit  of  the 
measuring  arc.  Draw  the  radius  vector  PB  equal  to  the 
given  base;  A  and  B  will  be  points  of  the  curve.  Other 
points  may  be  found  as  already  described. 

That  part  of  the  curve  below  the  line  PA  corresponds  to 
the  negative  value  of  r',  and  for  that  we  have 

I 
a" 

in  which  when  r^^o,  7'  will  be  infinite.     Hence  the  curve  is 
unlimited  in  both  directions. 


SECTION   X 


ASYMPTOTES. 

(88)  An  asymptote  to  a  curve  is  a  line,  which  the  curve 
continually  approaches,  but  never  meets.  Such  a  line  is 
said  to  be  tangent  to  the  curve  at  an  infinite  distance,  by 
which  we  are  to  understand  that  the  point  of  contact  to 
which  the  lines  approach  is  beyond  any  finite  limit. 

That  this  may  be  the  case  it  is  necessary  that,  at  least, 
one  of  the  coordinates  of  the  curve  may  have  an  unlimited 
value.  Hence  when  we  are  seeking  an  asymptote  to  a 
curve,  our  first  inquiry  must  be,  whether  the  equation  of  the 
curve  will  admit  of  such  values  for  the  coordinates  or  either 
of  them.  If  not,  there  can  be  no  asymptotes.  If  it  will  do 
so  for  either  coordinate,  we  must  substitute  that  value  in  the 
equation  and  ascertain  the  resulting  value  for  the  other 
coordinate.  If  this  resulting  value  is  finite,  there  is  an 
asymptote  parallel  to  the  axis  of  the  infinite  coordinate ;  if 
zero  then  the  axis  of  the  infinite  coordinate  is  itself  the 
asymptote.  But  if  it  should  be  infinite,  then  we  must  resort 
to  the  following  method. 

Find  from  the  equation  the  values  of  the  coordinates  at 
the  points  where  the  tangent  line  intersects  the  axes,  that  is, 

182 


ASYMPTOTES, 


187. 


the  distances  from  the  origin. 
These  points  may  be  found  as 
follows  : 

Let  A  (Fig.  37)  be  the  origin  of, 
coordinates  for  the  curve  SO,  and 
let  PB  be  tangent  to  the  curve  at 
the  point  P,  of  which  the  coordi- 
nates are  x'  and  y' .  The  equa- 
tion of  this  tangent  line  is  ^^s-  37- 

y-y  ~~j^'\.^-^ ) 

If  we  makejF=^  we  have 

dx 
■^  ay 
If  we  make  x^=^o  we  have 

If  EC  be  an  asymptote,  and  the  values  of  x  and  y  are 
made  such  as  to  remove  the  point  of  tangency  to  an  infinite 
distance,  then  AB  and  AD  will  become  AC  and  AE. 

If  in  such  case  we  have  finite  values  for  these  distances, 
then  there  will  be  one  or  more  asymptotes;  if  there  is  but 
one  finite  value,  there  will  be  one  asymptote  parallel  to  the 
axis  of  the  infinite  coordinate.  If  one  be  zero  then  the  axis 
of  the  infinite  coordinate  is  it-self  the  asymptote.  If  both 
be  zero  then  the  asymptote  passes  through  the  origin;  but 
if  both  be  infinite  there  is  no  asymptote. 

EXAMPLES. 


Ex.  I.     The  equation  of  the  hyperbola  referred  to  its  cen- 
ter and  asymptotes  is 

-i'  =  M 

in  which  if  x  is   made   infinite  y  becomes  zero  ;  and  if  y  is 


184  DIFFERENTIAL    CALCULUS. 

made  infinite  x  becomes  zero ;  hence  both  axes  are  asymp- 
totes 

Ex   2.     If  we   consider  the  hyperbola  as  referred  to  its 
center  and  axes,  its  equation  is 

where  either  x  ox  y  may  be  made  infinite,  and  such  value 
makes  the  other  infinite  also.  Hence  we  take  the  formulas 
for  the  points  of  intersection  of  the  tangent  with  the  axes^ 


which  give 

r       ,AV. 

A2j;'2_b2^t'"_A2 

x—x    y^2^'- 

B^ct'               x 

and 

,        ,B^V 

A2y2_B2^^'2               ]32 

7=7 -^AV  = 

^  y               y 

both  of  which  values  becomes  zero,  when.r'  andj'  are  made 
infinite.     Hence  the  asymptotes  pass  through  the  origin. 

Ex.  3.     The  equation  of  the  parabola 

j;  2  =  2px 

shows  that  x  and  j^  both  become  infinite  together,  and  hence 
we  take 


and 


_ ,    ,'^y  _  ,  p ,  _y 

y — y  — X  —r-,  — y  — —x   — 

•^     ^  ax  y  2 


both  of  which  values  become  infinite  when  X    and  y  are 
infinite,  and  hence  there  is  no  asymptote  to  the  parabola. 

Ex.  4.     If  we  take  the  ellipse  whose  equation  is 
A'-y-^B^x-^X^B" 
we  see  that  neither  x  nor  y  can  ever  be  infinite  ;  in  fact  y 
can   never  exceed  B   nor  x  exceed  A ;    hence  there  is  no 
asymptote  to  the  ellipse. 

Ex.  5.     The  equation  of  the  logarithmic  curve  is 

x^Xog.y 


ASYMPTOTES. 


It  may  be  constructed  by  laying  off  on  the  axis  of  abscis- 
sas (Fig.  38)  the  distances  AB,  AC,  AD,  etc.,  in  arithmeti- 
cal progression,  and,  on  the 
corresponding  ordinates,  the 
distances  Ka,  B(^,  C^,  Tyd,  etc., 
in  geometrical  progression, 
and  drawing  a  curve  through 
the  points  thus  found.  We 
see  from  the  equation  that  if 
either  x  or  y  is  infinite  on  the 
positive  side,  the  other  will  be 
infinite  also. 


\^    Z   F 


ABC 

Fig.  38. 

If  we  apply  the  formula  for  the  intersection  of 
the  tangent  line  with  the  axis  we  have  (Art.  t,S) 

y^y -""d^^y -^  u^y ^^- m i 

and 

x=^x  ^y  ~^'^^x  —y 


=jc'  —  M 


(<) 


ay  -^   y  ^   ^ 

We  see  from  these  values,  that  when  x'  is  infinite  x  will 
be  infinite  positively,  and  y  negatively.  Hence  there  is  no 
asymptote  on  the  positive  side  of  x.  But  if  x'  be  made 
infinite  negatively,  y  will  become  zero  ;  for  the  logarithm  of 
0  is  negative  infinity,  which  shows  that  the  axis  of  abscissas 
is  an  asymptote  on  the  negative  side.  The  value  of  y  how- 
ever in  equation  (i)  becomes  — 00c,  which  is  indefinite. 

We  learn  from  equation  (2)  that  the  tangent  always  inter- 
sects the  axis  of  abscissas  at  a  distance  equal  to  M  on  the 
negative  side  of  the  ordinate  of  the  point  of  tangency. 
Hence  the  subtangent  is  constant  and  equal  to  the  modulus 
of  the  system  to  which  the  curve  belongs.  If  x  =^l,  then 
X  and  y  both  become  zero,  and  the  tangent  passes  through 
the  origin. 

If  we  put  the  equation  into  the  form 


l86  DIFFERENTIAL    CALCULUS. 

and  make  x  negative  it  becomes 

which  makes  y=^c?  when  ^=00;  whence  we  infer  that  the 
axis  of  abscissas  is  an  asymptote  to  the  curve  on  the  nega- 
tive side,  as  already  shown. 


SECTION   XI. 


SIGNIFICATION   OF    THE    SECOND  DIFFERENTIAL 
COEFFICIENT. 


SIGN    OF    THE    SECOND    DIFFERENTIAL    COEFFICIENT. 

(89)  We  have  seen  (Art.  36)  that  the  first  differential  of 
the  ordinate  indicates  by  its  sign  whether  tlie  curve  is  leav- 
ing or  approaching  the  axis  of  abscissas ;  and  by  its  value  it 
determines  the  rate  of  such  approach  or  departure ;  that  is, 
the  tangent  of  the  angle  made  by  the  tangent  line  with  the 
axis  of  abscissas. 

As  the  point  of  tangency  moves  along  the  curve,  the  rate 
of  its  approach  to,  or  departure  from,  the  axis  of  abscissas 
is  constantly  changing,  and  upon  the  rate  of  this  change  will 
depend  the  direction  and  amount  of  curvature  of  the  curve. 

Wherever  the  curve  is  situated  with  reference  to  the  axis 
of  abscissas,  if  its  rate  of  departure  is  an  increasing  rate,  or 
its  rate  of  approach  is  a  decreasing  rate,  then  the  curve  is 
co7ivex  toward  the  axis  of  abscissas  ;  while  if  its  rate  of 
departure  is  decreasing,  or  its  rate  of  approach  is  increasing, 
it  will  be  concave  toward  that  axis, 

(90)  Now  the  second  differential  of  the  ordinate  will 
determine  by  its  sign  whether  the  first  is  an  increasing  or 
decreasing  function.  If  the  latter  is  positive  and  increas- 
ing, or  negative   and  decreasing,  its  rate  of  change  (that  is 

187 


1 88 


DIFFERENTIAL     CALCULUS. 


the  second  differential  of  the  ordinate)  will  be  positive  (Art.. 
3)  ;  but  if  it  is  positive  and  decreasing,  or  negative  and 
increasing,  its  rate  of  change  is  negative. 

Note. —  It  will  be  remembered  that  the  sign  of  the  differential  and  that  of  its  coeffi- 
cient are  always  the  same,  since  the  differential  of  the  independent  variable  is  always 
uniform  and  positive. 

(91)  If,  therefore,  the  second  differential  coefficient  should 
be  positive,  the  first  must  be  either  an  increasing  positive  of 
a  decreasing  negative  function  (Art.  3).  If  the  curve  is  on 
the  positive  side  of  the  axis  of  abscissas,  it  is  convex  to  that 
axis  ;  if  on  the  negative  side  it  is  concave. 

(92)  If  the  second  differential  coefficient  is  negative,  the 
first  must  be  either  an  increasing  negative  function,  or  a  de- 
creasing positive  one.  Hence  the  curve,  if  on  the  positive 
side  of  the  axis  of  abscissas  will  be  concave,  and  on  the 
negative  side  convex  to  that  axis. 

(93)  To  illustrate  these  principles  let  us  suppose  the 
second  differential  coefficient  to  be  positive,  then  the  first 
must  be  a  positive  increasing 
or  a  negative  decreasing  func- 
tion. The  curves  in  Fig.  40 
and  41  answ^er  to  these  con-  A 
ditions,  for  from  C  to  D  the 
first  differential  coefficient  is 
negative  (Art.  36)  and  de- 
creasing, while  from  D  to  E 
it  is  positive  and  increasing  in  both  cases. 

If  the  second  differential  coefficient  is  negative,  then  the 
first  must  be  positive  and  decreasing,  or  negative  and  in- 
creasing, and  we  find  the  carves  in  Fig.  39  and  42  to  answer 
these  conditions;  for  from  C  to  D  the  first  differential  coeffi- 
cient (Art.  36)  is  positive  and  decreasing,  while  from  D  to 
E  it  is  negative  and  increasing  in  both  cases. 

By  inspecting  these  figures  we  see  that  for  39  and  40  the 


SECOND    DIFFERENTIAL    COEFFICIENT.  189 

second  differential  coefficient  has  in  each  case  a  sign  con- 
trary to  that  of  the  ordinate,  and  that  both  curves  are  con- 
cave to  the  axis  AB  ;  while  in  curves  41  and  42  the  sign  is 
the  same  as  that  of  the  ordinate,  and  the  curves  convex  to 
the  axis.     Hence 

When  the  signs  of  the  second  differential  coefficient  and  of  the 
ordinate  are  contrary  to  each  other ^  the  curve  ivill  be  concave 
toward  the  axis  of  abscissas ;  wJien  these  signs  are  alike  the  curve 
zaill  be  convex  toiuard  that  axis. 

It  will  be  noticed  that  in  all  these  cases  the  first  differen- 
tial coefficient  changes  its  sign  at  D  where  it  becomes  zero, 
but  this  docs  not  affect  the  sign  nor  the  value  of  the  second 
differential,  for  the  first  may  be  changing  as  rapidly,  and  in 
either  direction  at  the  zero  point  as  at  any  other. 

(94)  To  illustrate  these  rules  let  us  take  the  general 
equation  of  the  circle 

{x-aY+{y-b)-=V.^ 
in  which  a  is  the  abscissa  and  b  the  ordinate  of  the  center. 

Differentiating  we  have 

,  Y 

dy  X — a 

dx  y  —  b 

and  -A 

cPy__         R- 
dx-  ~~{y-bY  Fig.  43. 

From  which  we  learn  that  so  long  as  y  is  greater  than  b 
the  second  differential  coefficient  will  be  negative,  while  it 
is  positive  where  y  is  less  than  /',  or  where  it  is  negative. 
We  see  also  from  the  figure  (Fig.  43)  that  above  the  line  DE 
where  _y  is  greater  than  b  the  curve  is  concave  toward  the 
axis  of  abscissas,  while  between  DE  and  the  axis  of  abscis- 
sas, where  y  is  positive  and  less  than  b,  the  curve  is  convex 
toward  that  axis.  Below  the  axis  of  abscissas  where  i'  is 
negative   the   second  differential  is  still   positive,  while  the 


190 


DIFFERENTIAL    CALCULUS. 


curve  is  concave  toward  the  axis.     All  of  which  corresponds 
with  the  rule. 

In  the  case  of  the  parabola  referred  to  its  vertex  and 
axis  we  have 

d^  y         J)^ 

a  fraction  whose  sign  is  always  contrary  to  that  of  yj  hence 
the  curve  is  always  concave  towards  the  axis  of  abscissas. 
The  same  may  be  said  of  the  ellipse  referred  to  its  center 
and  axes  from  whose  equation  we  have 

d^  y  B^ 

dx'^  Kry"^ 

In  the  case  of  the  hyperbola  referred  to  its  center  and 
asymptotes  we  have 

d"^  y      2y 
dx^      x^ 

a  fraction  whose    sign    is   always  the   same   as   that   of  y. 
Hence  the  curve  is  everywhere  convex  toward  the  axis. 

VALUE    OF    THE    SECOND    DIFFERENTIAL    COEFFICIENT. 

(95)  The  curvature  of  a  curve  at  any  point  is  the  ten- 
deiicy  of  the  tangent  line  at  that  point  to  change  its  direc- 
tion, as  the  point  of  tangency  is  moving  along  the  curve,  in 
obedience  to  the  law  of  chajige  derived  from  the  conditions 
which  govern  the  movement  of  the  generating  point. 

Note. —  The  curvature  then  ofa  curve  is  not  '^  its  deviation  from  the  tangent,"*  nor 
"its  departure  from  the  tangent  drawn  to  the  curve  at  that  point, "t  nor  is  it  "the 
angular  space  between  the  curve  and  its  tangent,":}:  nor  is  it  any  actual  change  in  the 
direction  of  the  tangent  line  as  the  point  of  tangency  moves  along  the  curve  ;  nor  does 
it  depend  on  any  such  change,  but  upon  the  law  which  governs  the  movement  of  the 
generating  point ;  for  it  is  this  law  which  fixes  the  te?tdency  of  the  tangent  to  change 
its  direction  and  this  tendency  is  the  curvature.  Hence  in  estimating  the  curvature  of 
a  curve  at  any  point,  we  consider  that  point  alone  and  seek,  7iot  any  actual  movement 
of  the  generating  point,  but  the  Ia7v  which  cotitrols  it. 

*Loomis.     tDavies.     ^^Church. 


SECOND    DIFFERENTIAL    COEFFICIENT.  191 

Hence  if  several  curves  as  CD,  CD',  C"D"  (Fig.  44)  have 
coincident  tangeiits  AB  at  the  point  A,  and  if  we  suppose 
the  point  of  tangency  to  be  at  any  instant  moving  alon^';  the 
curve,  carrying  with  it  its  own  tangent 
line,  that  one  whose  tangent  line  at 
the  moment  of  coincidence  is  chang- 
ing its  direction  most  rapidly  will 
have  the  greatest  curvature  at  that 
point.      For  the  i-ate  of  change  in  the  ^^s-  44- 

direction  of  the  tangent  is   the  measure  of  its  tendcficy  to 
change. 

Since  the  first  differential  coefficient  indicates  the  direc- 
tion of  the  tangent  to  a  curve,  by  means  of  the  tangent  of 
the  angle  made  by  it  with  the  axis  of  abscissas ;  the  second, 
which  is  simply  the  rate  of  change  in  the  first,  will  indicate 
the  rate  at  which  the  tangent  of  that  angle  is  changing  its 
value.  Now  as  between  two  curves  at  a  common  tangent 
point,  that  curve  in  which  the  tangent  line  tends  to  change 
its  dii-ection  most  rapidly,  will  be  the  one  in  which  the 
tangent  of  the  angle  made  by  that  line  with  the  axis  of 
abscissas  will  also  tend  to  change  its  value  most  rapidly,  and 
will,  therefore,  have  the  greatest  curvature,  while  if  these 
tendencies  are  equal  the  curvatures  are  equal,  and  this 
will  be  indicated  by  the  equality  of  the  second  differential 
coefficients. 


SECTION   XII 


CURVATURE    OF  LINES. 


THEOREM. 


(96)  The  curvatures  of  different  circles  are  inversely  propor- 
tional to  their  radii. 

The  curvature  of  a  circle  is  the  same  at  all  points  of  the. 
circumference,  and  all  circles  having  the  same  radii  have 
the  same  curvature. 

Since  the  change  in  the  direction  of  the  tangent,  as  the 
point  of  tangency  moves  around  the  curve  is  constant,  its 
actual  change  of  direction  for  any  given  movement  of  the 
point  of  tangency,  will  always  be  in  proportion  to  its  ten- 
deficy  to  change,  multiplied  by  the  length  of  the  arc  over 
which  the  movement  is  made,  and  may,  therefore,  be  repre- 
sented by  that  product ;  and  hence  the  tendency  to  change  or 
curvature  will  be  equal  to  the  actual  change  divided  by  the 
length  of  the  arc. 

Now  the  change  in  the  direction  of  the  tangent  is  equal 
to  the  angle  contained  between  its  two  positions,  which  is 
the  same  as  that  contained  between  the  two  radii  drawn  to 
the  extremities  of  the  arc.  Calling  this  angle  v  and  the 
length  of  the  arc  ^,  we  shall  have 

V 

curvature  =~ 
a 

192 


CURVATURE    OF    LINES.  193 

If  now  we  have  two  circles,  which  we  will  call  o  and  o\ 
whose  radii  are  r  and  r',  and  the  angles  at  the  center  for  the 
same  length  of  arc  a  are  v  and  v  ^  we  shall  have 

V 

curvature  of  o~—— 
a 

7' 

curvature  of  o'  =~ 
a 


curvature  of  o  :  curvature  of  o'  wv'.v  (i) 

v.  360°  \\  a\  2r.r 
V  :  360    :  -.a:  2T.r 


hence 

but 

and 

whence 
or 

f  r 

v'.v  :  '.  r   '.r 

Substituting  this  ratio  in  proportion  (i)  we  have 
curvature  of  o :  curvature  of  o  wr  \r 

Q.   E.   D. 

CONTACT    OF    CURVES. 

(97)  When  two  curves  have  a  common  point,  the  coordi- 
nates of  that  point  must  satisfy  both  their  equations.  This 
will  generally  be  a  point  of  intersection^  and  not  a  point  of 
contact ;  and  is  all  that  can  be  secured  by  having  but  one 
condition  common  to  the  two  curves. 

If  they  are  at  the  same  time  tangent  to  each  other,  at  the 
common  point,  then  another  common  condition  is  imposed 
and  there  is  a  contact  of  the  first  order. 

The  condition  required  in  this  case  is,  that,  for  the  point 
of  contact,  the  first  differential  coefficients  shall  be  the  same 
for  the  equations  of  both  curves.  For  since  the  curves  are 
tangent  to  each  other,  they  have  a  common  tangent  line,  and 


194  DIFFERENTIAL    CALCULUS. 

the  first  differential  coefficient,  which  determines  the  angle 
made  by  this  line  with  the  axis  of  abscissas,  must  be  the 
same  for  both  equations. 

If,  besides  this,  the  curves  are  required  to  have  the  same 
curvature  at  the  point  of  contact,  this  will  introduce  a  third 
condition,  which  is,  that  the  second  differential  coefficients 
shall  be  the  same  for  both  equations  (Art.  95). 

For  the  second  differential  is  the  rate  of  change  in  the 
first,  which  gives  the  direction  of  the  tangent  line,  and  the 
rate  of  change  in  this  direction  is  the  curvature.  This  is  a 
contact  of  the  second  order. 

If  now  it  is  required,  in  addition,  that  the  rate  of  change 
in  the  curvature  should  be  the  same  in  both  curves  at  the 
point  of  contact ;  we  must  introduce  a  fourth  condition, 
viz.,  that  the  //^//v/ differential  coefficient  should  be  the  same 
in  both  equations.  This  would  be  a  contact  of  the  third 
order.  And  thus  the  order  of  contact  would  become  higher 
for  every  new  condition  introduced  common  to  both  curves, 
and  every  new  agreement  between  the  successive  differential 
coefficients. 

If  then  we  wish  to  find  the  order  of  contact  of  two  given 
curves,  we  first  combine  their  equations,  and  determine  their 
common  point  if  they  have  one.  For  this  point  the  varia- 
bles will  have  the  same  value  in  both  equations.  If  the 
values  thus  found  being  substituted  in  the  first  differential 
coefficient  of  each  equation,  reduce  them  to  the  same  value, 
there  is  a  contact  of  the  first  order;  that  is,  they  have  a 
common  tangent  line  at  the  common  point. 

If  they  also  reduce  the  second  differential  coefficients  of 
the  two  equations  to  the  same  value  they  have  a  contact  of 
the  second  order,  and  so  on  for  the  successive  differential 
coefficients  ;  the  order  of  contact  being  determined  by  the 
number  of  coefficients  that  successively  become  equal  by 
the   substitution  of  the  values  of  the  common  coordinates. 


CURVATURE    OF    LINES.  19^ 

EXAMPLE, 

(98)  To  illustrate  this  rule  let  us  take  the  two  equations- 

and 

y2  —  2)'= 3  —  ^^  (2^ 

from  which  we  obtain  l)y  combination 

r  =  — I   and  x=o 
indicating  that  botli   the  curves   pass  through  the  point  of 
which  these  are  the  coordinates.     We  have  also  by  differen- 
tiating twice  —  for  equation  (i) 

dy       X  d "  y 


=—   and 


dx      2   ""'^  dx-       ^ 


and  for  equation  (2) 

dy  X  ^  (^~  y I  x^ 

dx  y — I    ''     ^    dx''^         y — i      (j'— i)^ 

Substituting  in  these  differential  coefficients  the  values  of 
X  and  V  just  found,  we  have  the  first  differential  coefficients 

dv       X  dy  X 

—J-  =■  ~=^o   and  "ir~  —  — ^o 

dx       2  dx         y —  I 

and  the  second  differential  coefficients 

d"y^  d'^  V I  x"^      ^ 

and        "  ~  ~ 


dx^       ^  dx'^  y — I      {y — i)-^      ^ 

from  which  we  infer  that  at  the  |)oint  whose  coordinates  are 
a:=o  andj'^ — i,  the  curves  have  a  contact  of  the  second 
order.  We  also  see  from  the  value  of  the  first  differential 
coefficient  that  at  that  point  the  tangent  to  both  curves  is 
parallel  to  the  axis  of  abscissas.  A  little  investigation 
would  show  that  the  first  curve  is  a  parabola,  and  the  sec- 
ond a  circle  tangent  to  the  first  at  its  vertex. 

(99)  The  constants  which  enter  into  the  equation  of  a 
curve  determine  the  conditions  which  govern  the  movement 
of  the  generating  point  for  that  kind  of  curve ;  which  must 
fulfil  as  many  conditions  as  it  has  constants.     Thus  the  cir- 


196  DIFFERENTIAL    CALCULUS. 

cle  whose  general  equation  contains  three  constants,  must 
fulfil  three  conditions,  namel}^  two  in  the  coordinates  of  the 
center,  and  one  in  the  length  of  the  radius.  The  ellipse 
must  fulfil  four  conditions,  namely,  the  coordinates  of  the 
center  and  the  lengths  of  the  two  axes. 

(100)  Now  if  one  curve  be  giveJi  complete  by  its  equation 
with  fixed  values  for  its  constants,  and  another  with  con- 
stants which  are  indeterminate,  and  capable  of  being  adjusted 
to  any  given  conditions,  we  may  easily  assign  such  values  to 
them  as  will  cause  the  curve  to  fulfil  such  conditions  as 
may  be  required  of  it.  We  may,  for  instance,  require  the 
curve  to  pass  through  a  given  point  in  a  given  curve.  This 
will  require  that  the  same  variable  coordinates  shall  satisfy 
the  equations  of  both  curves  for  that  point.  We  may  also 
require  them  to  have  a  common  tangent  at  that  point ;  this 
will  require  the  constants  to  be  so  adjusted  that  the  first 
differential  coefficients  of  the  two  equations  shall  be  equal. 
If  there  are  three  or  more  constants  in  each  equation  we  may 
require  such  values  as  will  cause  the  second  differential 
coefficients  to  become  equal  also,  thus  producing  an  equality 
of  curvature,  or  a  contact  of  the  second  order,  at  the  com- 
mon point.  And  thus  we  may  continue  until  the  order  of 
contact  is  one  less  than  the  number  of  constants  to  be  dis- 
posed of. 

(101)  In  order  to  make  this  adaptation  of  the  second  curve 
to  the  first  we  must  consider  its  constants,  or  as  many  of 
them  as  will  be  required  for  the  purpose  as  unknown  quan- 
tities (Art.  4)  and  construct  as  many  equations  as  may  be 
required  to  determine  them. 

These  equations  are  derived  from  the  conditions  to  be 
fulfilled  by  the  constants.  Thus  the  first  which  requires 
that  the  second  curve  shall  pass  through  a  point  of  the  first 
will  generally  be  met  by  the  proper  adjustment  of  a  single 
constant ;  and  an  equation  formed  by  substituting  in  that  of 


CURVATURE     OF    LINES.  I97 

the  curve  to  be  adjusted  the  vaUies  of  the  coordinates  of  the 
designated  point,  and  also  the  values  of  the  known  con- 
stants, will  determine  the  value  of  the  unknown  constant. 

If  it  is  required  that  the  two  curves  be  tangent  to  each 
other,  we  must  adapt  the  values  of  two  constants  to  this 
condition,  and  this  is  done  by  substituting  the  same  values 
of  the  common  coordinates,  and  of  the  remaining  constants 
in  the  j^/'i'/ differential  coefficients  of  the  two  equations,  and 
placing  them  equal  to  each  other,  thus  forming  a  second 
equation.  A  contact  of  the  second  order  may  be  secured 
by  fixing  the  value  of  a  third  constant  in  a  similar  way  by 
means  of  the  second  differential  coefficients  of  the  two  equa- 
tions. 

The  values  of  these  cojtstants  thus  determined  being  substituted 
in  the  gejicral  equation  of  the  requir-ed  curve^  will  produce  an 
equation  of  one  that  will  fulfil  the  required  conditions ;  that  is, 
one  that  will  intersect  at  a  given  point,  or  have  a  contact  of 
a  required  order. 

EXAMPLE. 

(102)  To  illustrate  these  principles  let  us  take  the  equa- 
tion of  the  ellipse  referred  to  its  center  and  axes 

A2j^-+B^Jt:2=A-B2 
and  the  general  equation  of  the  circle 

{x-aY^^y-bY^^^  (i) 

in  which  the  constants  are  arbitrary  and  may  be  adapted  to 
any  prescribed  conditions.  Suppose  we  say  that  the  cir- 
cumference shall  pass  through  the  upper  extremity  of  the 
conjugate  axis  where 

x^=-o  and  j-^B 

This  being  but  one  condition  will  require  the  adaptation 
of  but  one  constant.  Let  this  be  a^  while  we  make  R=A 
and  b=^o. 


198  DIFFERENTIAL    CALCULUS. 

Then  substituting  these  values  in  equation  (i)  we  have 

or 

whence 

a=±VA^-B^ 
and  the  equation  of  the  circle  becomes 

the  center  being  in  one  of  the  foci  —  the  plus  value  of  the 
radical  corresponding  with  the  focus  on  the  positive  side  of 
the  center. 

If  we  add  another  condition,  namely,  that  the  curves  shall 
be  tangent  to  each  other  at  the  same  point,  we  must  adapt 
the  value  of  f7ao  constants  to  these  two  conditions.  Let 
these  constants  be  a  and  /?,  and  make  R  =  2B.  Then  we 
must  construct  an  equation  between  the  first  differential 
coefficients  of  the  curves  ;  that  is 

B-.T      x—a 

Substituting  the  values  of  x  and  y  as  before  we  have 

B^o       o—a 

hence 

and  substituting  these  values  in  equation  (i),  we  have 

(B-/;)2=4B2 
Vv^hence 

/^=-B 

and  the  equation  of  the  required  circle  becomes 
^2-f(_y+B)2=4B2 

the  center  being  at  the  lower  extremity  of  the  conjugate 
axis  where  a^o  and  <^=— B. 

If  now  we  add  a  still  further  condition  there  shall  be  a 
contact  of  the  second  order  at  the  same  point  we  must  adapt 


CURVATURE    OF    LINES.  199 

the  values  of  tJiree  constants  to  that  condition,  by  forming  a 
third  equation,  between  the  second  differential  coefficients, 
thus 

^^ y'^~  y-b   ~     y-b  ^^^ 

Substituting,  as  before,  the  values  of  x^o  and  j=B  in 
equations  (i),  (2),  (3),  we  have  three  equations  from  which 
to  determine  the  values  of  the  three  constants;  thus 

B~^       o — a 
An]^BW^ 

B^    __       (B-^)^ 
A2B3-      B-^ 

From  the  second  we  obtain 

From  the  third  we  have 

B^-AS 


and  substituting  these  values  in  the  first  we  obtain 


B 
t] 
A2 


and  the  equation  of  the  circle  becomes 

oc'-\-{y—^)  =^0 

the  radius  being  equal  to  half  the  parameter  of  the  conju- 
gate axis  of  the  ellipse,  and  the  center  being  in  that  axis 
prolonged  in  a  negative  direction. 

(103)  In  this  last  case  we  have  the  highest  order  of  con- 
tact of  which  the  circle  is  capable,  and  hence  the  circle  is 
called  the  osculatrix  to  the  ellipse;  or  is  said  to  be  oscula- 
tory  to  it. 

An  osculatrix  to  a  curve  is  one  luhich  ha^  the  highest  order  of 


200  DIFFERENTIAL    CALCULUS. 

cojitact  with  it^  that  any  airve  of  the  same  kind  as  the  osculatrix 
can  have. 

Since  the  number  of  constants  limits  the  number  of  con- 
ditions that  can  be  assigned  to  a  curve,  and  since  the  pass- 
ing of  the  curves  through  the  same  point  is  one  condition, 
the  order  of  contact  can  only  be  equal  to  the  remaining 
number  of  possible  conditions  ;  namely,  the  number  of  con- 
stants, less  one,  which  enter  into  the  general  equation  ;  and 
this  will  be  the  same  as  the  order  of  its  highest  differential. 

EXAMPLES. 

(104)  Ex.  I.  To  find  the  equation  of  the  circle  oscula- 
tory  to  the   parabola,  whose  equation  is 

y^=^x  (i) 

at  the  point  where  the  coordinates  are 

-x^i  and  y^=^2. 
Differentiating  this  equation  we  have 


whence 


and 


iSi— _r       A  d^ y _      4 

7    —      anQ     ,  o  —        Q 

dx      y  dx'^  y'^ 


2  X — a 


y         y—b 


" /7\   2 


fx — a\ 


or 


and 


y^  y — b 

i—a 


/\—a\ 


(3) 


2-b 


CURVATURE    OF    LINES.  .  20I 

Also  from  the  general  equation  of  the  circle  we  have 

(l-^)2+(2— <^)2=R2  (i) 

and  from  these  we  find 

I^2z=32,      ^=5,      b—  —  2 
and  the  equation  of  the  circle  osculatory  to  the   parabola  at 
the  given  point  is 

(^-5)2+(j  +  2)~=32 

Ex.  2.     To  find  the  circle  osculatory  to  an   equilateral 
hyperbola  whose  equation  is 

at  a  point  whose  coordinates  are 

_>'— 4  and  .t=2. 
By  differentiating  we  have 

dy  _      r_ 
dx  X 

and 

d'^  y      2y 
dx^       x^ 
and  from  the  general  equation  of  the  circle  we  have 

(2-^7)2 +(4-/^)2  =R2  (j) 

2  — a 

7  =  2  (2) 

'  +  ^4-^J  (3) 

from  which  we  obtain 


R2=^'   a--^   1;=-^ 
4  '  2 


and  the  equation  of  the  required  circle  will  be 

(.T-7)-+0— -^j    =-j 

Ex.  3.     Find  the  equation  of  the  circle  osculatory  to  the 
curve  whose  equation  is 


202  DIFFERENTIAL    CALCULUS. 

4.y^=x^ — 4 
at  a  point  whose  coordinates  are 

x=o  y=^  —  I 

RADIUS    OF    CURVATURE. 

(105)  Since  the  curvature  of  a  curve  at  any  point  is  the 
same  as  that  of  its  osculatory  circle  at  that  point,  we  call 
the  radius  of  the  osculatory  circle  the  radius  of  curvature  of 
the  curve.  And  since  the  formulas  for  the  equation  of  the 
osculatory  circle  may  be  applied  to  any  point  of  a  given 
curve,  we  may  consider  them  as  expressing  the  general  con- 
ditions required  of  the  osculatory  circle. 

These  formulas,  as  we  have  seen,  are 

(^-^)2+(j;-/;)2=R3  (i) 

dy  X — a 


dx  y—b 

dy'^ 
d^y         ^~^~d^ 


dx^  y—b 

the  two  last  may  be  written 


(?) 
(3) 


•^-^=— ;^-(v-^)  (2) 

and 

dx"-{-dy'^  .  . 

If  we  represent  the  coordinates  of  any  given  point  in  a 

curve  by  x   and  y\  then  for  the  osculatory  circle  we  must 

have 

_   ,       _  ,     dy  _dy       d^ y _d"y 
x=x  ,  y—y  ,  -^--^y    -;^--^ 

The  quantities  a  and  b  represent  the  coordinates  of  the 
center  of  the  osculatory  circle,  and  R  is  its  radius. 

If  we  substitute  in  equation  (2)  the  value  o^  y—b,  we 
have 


CURVATURE    OF    LINES.  203 


ay  (ax-  -\-dy''\ 
whence  equation  (i)  becomes 


Ti^A      d-^y     )    "^V     d-^y      )    ~^ 
fro;n  which  we  have 

•  (dx^+dy^)^ 

^~^       dxd'^y  ^^^ 

which  is  the  general  expression  for  the  value  of  the  radius  of 
curvature  i7i  terms  of  quantities  belonging  to  a  given  curve. 

If  we  denote  the  length  of  the  curve  by  u  we  shall  have 

_    du^ 
dxd  '^y 

(106)  Since  the  curve  and  its  osculatory  circle  have  a 
common  tangent,  they  will  also  have  a  common  normal ; 
and  as  the  normal  to  the  circle  passes  through  the  center, 
the  normal  to  any  curve  at  any  point  will  pass  through  the 
center  of  the  circle  osculatory  to  it  at  that  point. 

This  is  also  shown  from  equation  (2)  which  is 

dy  x  —  a 

dx  y  —  b 

X  and  y  being;  coordinates  both  to  the  given  curve  and   to 

the  osculatory  circle  at  the  point  of  contact,  and  a  and  b  the 

coordinates  of  the  center  of  the  circle. 

For  since  "tt  is  the  tangent  of  the  angle  made  by  the 

tangent  line  with  the  axis  of  abscissas,  we  shall  have 

dx y—b 

dy      x—a 
for  the  tangent  of  the  angle  made  by  the  normal  line  with 
the  same  axis.     But  when  a  straight  line  passes  through  two 
points  —  X  and  y  being  the  coordinates  of  one,  and  a  and  b 
the   coordinates  of  the   other — the  tangent   of   the   angle 


204  DIFFERENTIAL    CALCUL-UTa. 

made    by    that    line   with    the    axis    of    abscissas    will    be 

y — ^ 

,  and  hence  the  normal  to  the  curve,  since  it  passes 

through  the  first  point  will  also  pass  through  the  second  — 
that  is,  the  center  of  the  osculatory  circle. 
And  since  from  equation  (3)  we  have 


(y-^^'^-(^-^%) 


the  value  of  the  first  member  of  the  equation  will  be  essentially 

a  "  y 
negative,  and  hence  we  infer  thatj^^— <^  and     ,  3    must  have 

contrary  signs.     So  that  if     ,  g    is  negative,  d  will  be  less 

than  y,  and,  if  positive,  it  will  be  greater.  In  the  first  case 
the  curve  will  be  concave  toward  the  axis  of  abscissas,  and 
<^  will  be  between  the  curve  and  that  axis ;  while  in  the 
other  case  the  curve  will  be  convex  toward  the  axis  of 
abscissas,  and  ^  will  be  beyond  it.  Hence  the  center  of  the 
osculatory  circle  will  be  on  the  concave  side  of  the  curve. 
(107)  To  find  the  general  expression  for  the  radius  of 
curvature  of  the  parabola,  we  differentiate  its  equation  twice 

and  obtain 

ydy  ^^pdx 
and 

whence 


yd'^y-\-dy^'=o 
pdx 


dy^'- 


and 


y 


dy^  p^dx'"' 


y  y 

Substituting  these  values  in  the  formula  we  have 


(^dx^ 


p^dx^J 


3  3 


_  y^    J        [dx^y^-^p^)y  _(2px+p^Y 

^"~      -dx^-^  ~         -p^dx^        ~       -p^ 

y6 


CURVATURE    OF    LINES.  205 

or,  the  cube  of  the  normal  (Art.  56)  divided  by  the  square 
of  half  tlie  parameter. 

If  we  make  x^o  we  have 

or  half  the  parameter  for  the  radius  of  curvature  at  the 
vertex.     If  we  make  oc^\p  we  have 

for  the  radius  of  curvature  at  the  point  where  the  ordinate 
through  the  focus  meets  the  curve.  As  every  other  value 
of  R  is  greater  than  that  where  .t=o  it  follows  that  the 
greatest  curvature  of  the  parabola  is  at  the  vertex. 

(108)  From  the  equation  of  the  circle  we  have 

dy^- 

y 

and 

and  substituting  these  values  in  the  formula  we  have 

the  radius  of  the  circle  as  it  should  be. 

(109)  From  the  equation  of  the  ellipse  we  have 

W-xdx 

and 

A2,/v2_|_B2,/^2 

d^  y=^  — TT^ 

or  substituting  in  the  last  equation  the  value  o(  dy^  we  have 


d^y- 


A~y 


-1,3 


These  values  being  substituted  in  the  formula 


2o6  DIFFERENTIAL    CALCULUS. 


B*./Jt:3  B 


A2^3  A- 

which   is  equal  to  the  cube  of  the  normal  divided  by  the 
square  of  half  the  parameter  as  in  the  parabola. 
If  we  make  x=^A  we  havej^=o  and 

B3 

If  jF=B  then  x=^o,  and  we  have 

A2 
^  =  -B 
Hence  the  radius  of  curvature  of  the  ellipse  at  the  princi- 
pal vertex  is  half  the  parameter  o^  the  transverse  axis  — 
that  is  the  ordinate  through  the  focus.  At  the  vertex  of  the 
conjugate  axis,  the  radius  is  half  the  parameter  of  that  axis 
(Art.  102). 

(110)  The  equation  of  the  hyperbola  referred  to  its  center 
and  asymptotes  gives 


X 


and 


2dxdy 
d-y  —  — 


X 

Substituting  these    values   in   the  formula  we    have    after 
reducing. 

R  = 


(:r^+7^)^      2(x^  -\-y^)^ 
2xy  A^+B" 

In  the  equilateral  hyperbola,  this  value  becomes  equal  to 
the  cube  of  the  radius  vector  divided  by  the  square  of  the 
semi-axis. 


SECTION    XIII. 


E  VOLUTES. 

(111)  If  we  suppose  a  circle  to  roll  along  the  concave 
side  of  a  curve,  being  always  tangent  to  it,  and  at  the  same 
time  varying  the  length  of  its  radius  so  as  to  be  osculatory 
also,  its  center  will  describe  a  curve  which  is  called  the 
evolute  of  the  given  curve  ;  and  its  variables  will  be  the 
coordinates  of  that  variable  center.  In  other  words,  the 
evolute  of  any  curve  is  the  locus  of  the  centers  of  all  the 
circles  that  can  be  drawn  osculatory  to  that  curve. 

The  relation  between  the  variables  of  the  evolute  can  be 
determined  and  its  equation  found  from  the  equation  of  the 
given  curve,  and  the  first  and  second  differential  coefficients 
derived  from  that  equation ;  since  these  determine  the  posi- 
tion and  length  of  the  radius  of  curvature,  and  consequently 
the  place  of  the  center  of  the  osculatory  circle. 

Since  the  coordinates  of  the  point  of  tangency  and  the 
first  and  second  differential  coefficients  are  the  same  for  the 
given  curve  and  for  the  osculatory  circle,  we  can  at  once 
determine  two  of  the  properties  of  the  evolute. 

(112)  The  first  of  these  properties  is,  the  radius  of  the 
osculatory  circle  is  tangent  to  the    ^,JC 
evolute. 

Let  AC  (Fig,  45)  be  any  curve, 
and  let  c  be  the  center  of  the  oscula- 
tory circle  for  the  point  A,  while  c\ 
c'\  c"  are  the  centers  of  the  oscula- 
tory circles   corresponding  to  the 

2C7 


2o8  DIFFERENTIAL     CALCULUS. 

points  A',  A." ,  Ps!" .  Then  the  curve  cc"'  passing  through  these 
centers  will  be  the  evolute,  and  any  radius  as  AV  will  be  tan- 
gent to  it  at  the  point  c\  the  center  of  the  osculatory  circle. 
The  equations  of  conditions  (Art.  105)  may  be  put  into 
the  following  form 

{x-aY+{y-bY=R.^  (i) 

{x  —  a)dx-\-{y— l?)dy=-o  (2) 

{y—^)d^y-\-(^y^-{-i^x'^=^o  (3) 

and  in  this  case  a,  b,  R,  x,  y  are  variables ;  x  being  indepen- 
dent and  dx  a  constant  quantity ;  while  x  and  y  are  coor- 
dinates of  the  given  curve,  and  of  the  osculatory  circle  at 
the  point  of  contact,  and  a  and  b  coordinates  of  the  varia- 
ble center  of  the  osculatory  circle,  that  is,  of  the  evolute, 
and  are  functions  of  x  and  j. 

From  these  equations,  as  we  have  seen  (Art.  105),  R  may 
be  determined  for  any  point  in  the  given  curve  by  eliminating 
a  and  b  considered  as  constants.  But  for  the  evolute  curve 
we  must  consider  them  as  variable  coordinates  ;  and  hence 
under  that  supposition  if  we  differentiate  equations  (i)  and 
(2)  we  have 

{x—d)dx-\-{y—b)dy—{x—a)da'~{y—b)db  =  ^dV^      (4) 
and 

dx^  -\-dy^  -\-{y—b)d^y  —  da  .dx—db.  dy=o  (5) 

Subtracting  equation  (2)  from  (4)  we  have 

-{x-a)da-{y-b)db  =  KdK  (6) 

and  subtracting  (3)  from  (5)  we  have 

—  da  .  dx—db  .  dy^=-o  (7) 

whence 

db  dx 

da  dy  (8) 

dx  . 
but    ——r-  IS  the  tangent  of  the  angle  made  by  the  normal 

line  to  the  curve,  at  the  point  whose  coordinates  are  x  and 7. 

db  . 
with  the  axis  of  abscissas;  and  ^r-is  the  tang-entof  the  an- 

'  da  ^ 


EVULUTES.  209 

gle  made  by  the  tangent  line  to  the  curve,  at  the  point  whose 
coordinates  are  a  and  ^,  with  the  same  axis.  But  x  and  y 
are  coordinates  of  the  given  curve,  which  is  now  called  the 
involute^  and  a  and  b  are  coordinates  of  the  e volute,  and,  of 
course,  of  the  center  of  the  osculatory  circle  corresponding 
to  the  point  [^x.y]  on  the  curve,  and  through  this  center 
the  normal  line  must  pass  (Art.  106);  2nd  since  both  the 
normal  to  the  curve  (or  radius  of  curvature)  and  the  tan- 
gent to  the  evolute  pass  through  the  same  point,  and  make 
the  same  angle  with  the  axis  of  abscissas,  they  must  be 
one  and  the  same  line;  and  hence  the  proposition. 
(113)  The  other  property  referred  to  in  Art.  1 1 1  is 
The  difference  behveen  the  len^^th  of  the  evolute  curve  and  the 
radius  of  cu7'vature  of  the  involute,  measured  fro?n  the  same 
pointy  is  either  zero  or  a  constant  quantity. 

From  equations  (2)  and  (8),  of  the  preceding  article,  we 
have 

x—a='-^'y  —  b)  (9) 

(to 

and  substituting  this  value  of  x—a  in  equation  (i)  we  have 

0'-^)='|-:+0-'")==R==0-^)^'^^'      (-) 

From  equations  (9)  and  (6)  we  have 

which  being  squared  gives 

{da^  -\-db-Y     P2VP2 

and  this  being  divided  by  equation  (10)  gives 

da'^-\-db^=d^^ 
If  we  designate  the  length  of  the  evolute  by  u  we  shall  have 

du'^da'^-^db'^ 

whence 

du'^^d^^ 


2IO  DIFFERENTIAL    CALCULUS. 

or 

du=^d^  or  <^R— ^//=o=^/(R— 2/) 
hence  R  —  ii  is  a  constant  quantity  and 

R=z^+^ 
If  u=^o  we  have 

R=^ 
and  hence  c  is  equal  to  the  radius  of  curvature  at  the  begin- 
ning of  the  curve,  and  R  is  at  all  times  equal  to  the  length 
of  the  evolute  to  the  point  where  R  is  tangent  plus  the  con- 
stant c. 

If,  therefore,  we  suppose  a  cora  to  be  fastened  at  B  (Fig. 
45)  and  drawn  tight  around  the  curve  AB  and  then  unwound 
from  A,  the  end  of  the  cord  will  describe  the  curve  AC  of 
which  the  curve  AB  is  the  evolute.  For  the  cord  will  be  at 
all  times  tangent  to  the  curve  from  which  it  is  unwound,  and 
also  the  momentary  radius  of  the  curve  AC  for  the  point  at 
its  own  extremity,  and  consequently  normal  to  the  curve  at 
that  point ;  while  the  length  of  the  cord  from  the  point  of 
tangency  to  its  extremity  in  the  curve  AC  is  equal  to  the 
distance  from  the  same  point  to  the  origin  at  A  measured 
along  the  curve  AB. 

(114)  To  find  the  equation  of  the  evolute,  we  must  com- 
bine the  equation  of  the  osculatory  circle  with  that  of  the 
involute  in  such  a  manner  that  x  .y  and  R  shall  disappear 
and  leave  an  equation  containing  only  a  and  b  as  variables. 

This  will  require  four  equations,  and  these  are  obtained 
from  the  equation  of  the  involute,  the  general  equation  of 
the  circle,  and  those  formed  by  placing  the  first  and  second 
differential  coefficients  of  each  of  these  equations  respec- 
tively equal. 

Thus  if  we  take  the  equations  of  condition  (Art.  105) 

(^-^)2+(j;-^)2=R3  (i) 

(fv ,        ,,  dv  x—a 


EVOLUTES.  211 

dx"^  -^dy^  d^  V  ^  "*"  dx^ 

y-^'^—^ny-   or  -^.^—yzT  (3) 

and  then  differentiating  the  equation  of  the  involute  t\vice> 
we  find  the  values  of  the  same  differential  coefficients  and 
make  them  equal  to  the  second  members  of  equations  (2) 
(3) ;  then  eliminate  x^y  and  R,  the  resulting  equation  is  that 
of  the  evolute. 

Since  R  is  contained  in  only  one  equation,  we  omit  that, 
as  the  remaining  three  are  sufficient  for  eliminating  x  and  y^. 
and  for  the  resulting  equation. 

(115)  To  find  the  equation  of  the  evolute  to  the  parabola. 

The  equation  of  the  parabola  is 

y-—2px  (i) 

from  which 

dx      y  ^   ^ 


.  —         ,  {A 

dx^  y^  ^^' 


and 

d^y_     p^ 

a 

Placing  these  differential  coefficients  equal  to  those  of  the 

general  equation  of  the  circle,  we  have 

/         X — a 


y         y—b 
and 

dy^ 


(4) 


i;3  y ^  VD/ 

Dividing  equation  (5)  by  equation  (4),  and  substituting  for 
y"^  its  value  from  equation  (i),  and  reducing,  we  have 

^=3.T+/  (6) 

and  substituting  the  values  of  a  and  y  in  equation  (4)  we 
have  after  reducing 

{2XY 


b^- 


P' 


212 


DIFFERENTIAL     CALCULUS. 


and  substituting  in  this  the  value  of  x  from  equation  (6), 
and  squaring,  we  have 


ia-p)- 


yp 


-^p^--PY 


which  is  the  equation  of  the  evolute  of  the  parabola. 

If  we  make  b^^o  we  have  <^=/,  which  is  the  center  of  the 
osculatory  circle  for  the  vertex.  If  we  transfer  the  origin  to 
that  point  we  have 

a^^p-\-d   and  b'=^b' 
hence 


27 

Since  every  value  of  a  gives  two 
equal  values  for  tl  with  contrary 
signs,  the  curve  of  the  evolute  ACE 
(Fig.  46)  is  symmetrical  about  the 
axis  of  abscissas.  If  a  is  negative  "^ 
then  b'  is  imaginary,  and  hence  the 
curve  commences  at  C,  a  point  in  the 
axis  of  abscissas  at  a  distance  from 
A  equal  to  /  —  that  is,  at  double  the 
distance  of  the  focus,  or  half  the  parameter. 

(116)  To  find  the  equation  of  the  e-volute  of  the 

For  this  case  we  have 

A2j;2  4-B2jc2=A2B2 


dx 


^^x 


x  —  a 


h.^y  y  —  b 


B^ 


dx^ 


h.^y' 


df 

y—b 


ellipse. 

{2) 
(3) 


From  equations  (2)  and  (3)  we  have 


y—b=^ 


A^y{x—a) 


-a)     AVHi+^f) 


B' 


EVOLUTES.  213 


whence 

whence 

whence 
whence 
whence 
whence 


A2:v(A2B3-A2j;2)=:A*B2^+B*^3 

A2B-^.v^=A*B2,?+B'ijc3 

A2-B- 


A 


x^  .  (4) 


Substituting  this  value  of  a  in  equation  (2)  we  have 

^-     A*     "^^~       A~y  . 

whence 

A^-(A^-B^).x2_(j'-/^)B2 

A^  ~      y 

whence 

AV-A2jt:'-j'  +  B2jt:2^=A2B2^-A2B2^ 

whence 

j'(A2-jc--j;2)  =  -B2^ 

Substituting  for  .t-  its  value  from  equation  (i)  we  have 

A-B2-A-v2 

whence 

A--B- 

^=—^r'y'  (5) 

flaking  A^— B"=^-  we  have 

2  /•2 


214  DIFFERENTIAL    CALCULUS. 


and  making  —r-'=m  and  -^"=^2,  we  have 

a       x^  b  y^ 

m      A^  n  B^ 

or 

4  J. 

^  V  f  b\^ 


Writing  the  equation  of  the  ellipse  under  the  form 

x'^       y^ 
X^  +  B^^' 

x       ,   y    . 

and  substituting  the  values  of  -7-  and  -rr-  just  found,  we 
have 

which  is  the  equation  of  the  evolute  of  the  ellipse  in  which 
a  and  b  are  the  variable  coordinates,  and  7/1  and  n  the  con- 
stants. If  we  make  a=o  we  have  b^=  +  7i,  and  if  b=o  we 
have  a=±7/i,  which  shows  that  the  form  of  the  evolute  is 
symmetrical  with  both  axes  of  the  ellipse.     But 

^3       A^-B^ 

and  subtracting  this  from  A  we  have  the  radius  of  curvature 

B2 

at  the  principal  vertex  equal  to  — r",  as  we  have  already  seen 

(Art.  109).     Similarly  we  find  the  radius  of  curvature  at  the 

vertex  of  the  conjugate  axis  to  be  ~^. 

If  we  differentiate  equation  (6)  twice  we  have 

"^      i/b\'~^^/b 

whence 


-(-)    4—  -)    -r=o 

m^vi'  11^  n'       da 


and 


EVOLUTES 

I  /   (7\      ^ 

db 

in  \  ///  / 

da  ~ 

^  (  b\    ^ 

7i\  n' 

>-i 

T     /  /;x~3',//,2 

215 


_i 


7/1^  b/n 


77i^^m'  ir\)ii      da-      ii\  uf       da^ 


whence 


d^b 

da^ 


\   f  a\    "^       \    (  o\    "u 


/b^ 


11^  11' 

Since  the  numerator  of  the  second  differential  coefficient 
is  always  positive,  it  will  have  the 
same  sign  as  the  denominator,  which 
is  the  same  as  that  of  b^  and  hence  A 
the  curve  is  everywhere  convex 
toward  the  axis  of  al)scissas.  The 
first  differential  coefficient  becomes 
zero  when  ^=6?,  and  infinite  when  <^=<?,  hence  both  axes 
are  tangent  to  the  curve,  as  in  Fig.  47. 

If  we  make  A=B,  then  <r=^,  and  also  in^^o  and  ;/=^, 
hence  a  and  b  in  equations  (4)  and  (5)  will  also  become 
zero  as  they  should,  since  in  case  of  the  circle  the  evolute 
is  reduced  to  a  point  — the  center. 


Fig-  47- 


SECTION   XIV 


ENVELOPES. 


C 
d 

dl\   Fiq,  48 


(!I7)  Suppose  two  lines,  AB  and  AC  (Fig.  48),  be  drawn 
at  right  angles  to  each  other,  and  a  third  line  ^^/  to  move  in 
such  a  manner  that  its  extremities  <-/  and  e  shall  be  con- 
stantly in  these  axes,  while  its  length  remains  unchanged  ^ 
so  that  while  the  extremity  e  arrives 
successively  at  the  points  e\  e\  the 
extremity  d  will  arrive  at  the  corres- 
ponding points  d" ^  d' . 

During  this  movement  those  points    j 
of  the  line  near  the  extremity  d  will 
move  in  the  direction    more   nearly 

parallel  to  the  axis  AC  than  the  line      ^-rn — e w — e^ 

itself  is,  and  will  consequently  fall  within  its  first  position, 
while  the  points  near  the  extremity  e  will  move  in  a  direc- 
tion more  nearly  parallel  to  the  axis  AB  than  the  line  is,  and 
will  consequently  fall  without \X.'s>  first  position.  But  between 
these  extreme  points  there  is  one  that  tends  to  move  in  the 
direction  of  the  line  itself. 

This  point  does  not,  of  course,  remain  fixed  on  the  line, 
but  moves  from  one  extremity  to  the  other  as  the  line  changes 
its  position  and  direction,  always  occupying  that  place  in  the 
line  which  at  the  moment  does  not  tend  to  move  out  of  it 

216 


ENVELOPES.  217 

towards  either  side.      The  curve  described  by  this  point  is  the 
envelope  of  the  ctcrve. 

Again  let  AB  (Fig.  49)  be  the  transverse  axis  of  an 
ellipse,  and  CD  its  conjugate  axis;  and  suppose  these  axes 
to  vary  to  any  extent  under  the  condition  that  the  area  of 
the  ellipse  shall  remain  constant. 

Then  as  AB  decreases  CD  will 
increase  at  a  rate  corresponding  with 
this  condition.  When  the  curve 
thus  commences  to  change  its  shape, 
a  point  near  the  extremity  A  will  ■      v    \  1 

tend   to  move  in   a  direction   more  ^"\L      ^ 

nearly  parallel  to  AB  than  the  tan-  V^l'/^ 

gent  to  the  curve  at  that  point  is ,  ID^^^ 

while  a  point  near  the  extremity  C  Fig.  49- 

will  tend  to  move  in  a  direction  more  nearly  parallel  with 
CD  than  the  corresponding  tangent  line  is.  Now  between 
A  and  C  there  is  a  point  in  the  curve  that  tends  (as  the 
axes  are  changing)  to  move  exactly  in  the  direction  of  the 
tangent  to  the  ellipse  at  that  point. 

As  the  curve  changes  its  shape  and  position  this  point 
will  also  change  its  place  on  the  ellipse,  keeping  always 
where  its  tendency  is  in  the  direction  of  the  tangent  to  the 
ellipse  as  it  is  at  the  moment.  The  movement  of  the  point 
will  be  continuous,  and  it  will  generate  a  curve  which  v/ill 
be  the  envelope  of  the  ellipse. 

(!!8)  Since  the  point  on  the  given  curve  which  describes 
the  envelope  always  tends  to  move  in  the  direction  of  the 
momentary  position  of  the  tangent  to  the  curve  at  that 
point,  and  since  any  generating  point  always  tends  to  move 
in  the  direction  of  the  tangent  to  its  own  curve,  it  follows 
that  the  given  curve  and  its  envelope  will  have  a  common 
tangent  line  wherever  the  generating  point  mny  l)e  at  the 
moment  durino;   the  formation  of  the  curve.     Thus  in  the 


2l8  DIFFERENTIAL    CALCULUS. 

last  illustration,  the  ellipse,  in  every  stage  of  its  change,  will 
be  tangent  to  the  envelope  at  that  point  of  the  curve  just 
then  generated. 

(119)  An  envelope  to  any  line,  is  another  line  generated  by  that 
point  of  the  given  line,  which  tends  to  inove  in  the  direction  of  the 
tangent,  whenever  its  position  or  shape  is  made  to  change  by  chang- 
ing the  constants  of  its  equation,  or  any  of  them,  into  variables. 

An  envelope  is  not  always  produced  by  this  change  of  the 
constants,  for  it  may  be  that  no  point  of  the  given  line  will 
tend  to  move  in  the  direction  of  its  tangent;  as  in  the  case 
of  an  ellipse  where  both  axes  are  increased. 

In  general,  there  will  be  an  envelope  only  where  the  suc- 
cessive positions  of  the  line  corresponding  with  minute 
changes  in  the  constants,  will  intersect  each  other ;  for  while 
the  generating  point  of  the  envelope  tends  to  move  in  the 
direction  of  the  tangent,  the  points  on  each  side  of  it  v/ill 
tend  to  move  away  from  the  tangent  in  opposite  directions, 
hence  the  next  position  of  the  changing  line  will  cross  the 
previous  one  near  the  generating  point  of  the  envelope. 

(120)  If  in  any  equation  of  a  line  the  constants  are  made 
to  vary  in  value,  it  is  evident  that  while  the  curve  or  line 
remains  the  same  in  kind,  its  shape  and  position  may  assume 
every  possible  form  and  place  within  the  limits  determined 
by  the  law  of  variation  imposed  upon  the  constants  of  the 
equation. 

If  we  take  for  example  the  ellipse,  and  consider  A  and  B 
in  its  equation  as  independent  variables,  then 

will  represent  an  infinite  number  of  ellipses  of  every  possi- 
ble size  and  proportions  subject  to  but  two  conditions ; 
namely,  the  axes  must  both  coincide  with  the  axes  of  coordi- 
nates. If  we  make  A  and  B  dependent  on  each  other  we 
limit  the  system  of  ellipses  by  the  condition  thus  introduced, 
but  still  their  number  is  infinite.     If  we  introduce  the  still 


ENVELOPES.  219 

further  condition  that  the  values  of  .r  and  y  shall  be  confined 
to  those  points  of  the  system  which  tend  to  move  in  the 
direction  of  the  tangent,  while  A  and  B  tend  to  change  their 
values,  the  first  differential  coefficient  will  not  be  affected  by 
such  tendency  in  A  and  B,  and  hence  will  be  the  same  af 
tho:e  points  whether  they  are  considered  as  variables  or  con- 
stants. So  then  if  we  take  the  differential  of  the  equation 
with  respect  to  them  only  as  variables,  and  make  it  equal 
to  zero,  and  i-ncorporate  it  with  the  original  equation,  we 
put  this  limit  on  the  values  of  x  and  7,  which  will  then 
only  apply  to  points  in  the  envelope.  The  equation  will, 
therefore,  be  that  of  the  envelope  itself — that  is,  instead  of 
representing  every  point  i?i  one  ellipse^  it  will  represent  07ie  point 
171  each  quadrant  of  every  ellipse  that  .can  be  fontied  tender  the 
given  conditions. 

To  find  the  equation  then  of  an  envelope  we  differentiate 
the  equation  of  the  given  line  with  reference  to  such  only 
of  the  constants  as  are  considered  variable  for  the  time 
being,  and  place  that  differential  equal  to  zero.  The  values 
of  the  constants  determined  from  this  equation,  and  the 
conditions  of  relation  among  themselves,  being  substituted 
in  the  given  equation,  will  produce  one  that  will  be  inde- 
pendent of  the  variable  constants,  and  this  will  be  the  equa- 
tion of  the  envelope. 

EXAMPLES. 

(121)  For  the  first  example,  let  us  take  the  general  equa- 
tion of  the  circle  in  which  R  and  b  are  constants,  while  a  is 
considered  as  a  variable.  Now  since  the  values  of  x  and  y 
are  to  be  confined  to  those  ]:)oints  of  the  circle  which  tend 
to  move  in  the  direction  of  the  tangent  while  a  varies,  it 
will  make  no  difference  whether  we  differentiate  with  re- 
spect to  ,v  and  ]'  only,  or  with  respect  to  a  also.  Differenti- 
atincr  in  both  these  wavs  we  have 


220 


DIFFERENTIAL    CALCULUS. 


(.T — a)(Lx  -\-{y—b)dy=o 


and 


(x—a)ifx—{x—d)da  +  (y—b)dy=^o 
making  these  differentials  equal,  and  cancelling  like  terms 
we  have 

—  (:x—a)da—o  (i) 

which  we  should  have  obtained  at  once  by  differentiating 
with  respect  to  a  alone,  considering  all  the  rest  as  constants. 
From  equation  (i)  we  have 

x'=-a 
and  this  value  substituted  in  the  general  equation  gives 
y—b—±^  or  j;=^±R 

If  we  take  the  positive  value  for  R,  this  is  the  equation 
of   a  line    DE    (Fig.    50) 
parallel  to  the  axis  of  ab- 
scissas at  a  distance  equal 
to  that  of  the  centers  of 
the  system  of  circles  plus 
the  radius,  and  hence  tan-   * 
gent   to  them   all  on   the       j 
upper  side,  and  is  genera-  ^'g-  5°- 

ted  by  the  highest  point  of  the  circle  as  it  moves  from  D  to 
E,  as  a  varies  in  value ;  that  point  being  the  one  that  tends 
to  move  (and  in  this  case  does  move)  in  the  direction  of  the 
tangent  to  the  circle  drawn  through  it.  If  we  take  the  neg- 
ative value  of  R,  the  equation  represents  the  line  D'E'  tan- 
gent to  the  system  of  circles  on  the  lower  side. 

(122)  If  we  take  the  same  equation  and  consider  a  and  // 
both  as  variables,  we  must  establish  a  relation  between  them 
in  order  to  make  them  both  functions  of  x  and  y.  Let  this 
relation  be  expressed  by  the  equation 

a"  -\-b~  ^c"  (i) 

then  the  two   equations  will  represent  a  system  of  circles 


ENVELOPES. 


221 


(Fig.  51)  whose  centers  lie  in  the 
circumference  of  another  circle 
whose  radius  is  equal  to  r,  and  its 
center  is  at  the  origin. 

Differentiating  the  general  equa- 
tion of  the  circle  with  respect  to  a  \ 
and  b  only  we  have 

whence  ^'g-  si- 

db  _     x—a 
da  y  —  b 

We  may  now  substitute  for  b  its  value  obtained  from  equa- 
tion (i);  or  we  may  consider  it  as  a  function  of  a  in  that 
equation  and  substitute  the  value  of  the  differential  coeffi- 
cient derived  from  it.     This  will  give  us 

db  a X — a 

da  b         y—b 

whence 


(^_,,)2^ 


b^ 


Substituting  this  value  of  (.r — rt:)-in  the  general  equation  of 
the  circle,  we  have 

{c^-b^){y-by 


b-' 


from  which  we  obtain 


and  similarly 


b=- 


-f(j;-^)2=R2 


cy 


r±R 


ex 


a — ■ 


r±R 

Substituting  these  values  of  a  and  b  in  the  general  equation 
of  the  circle  we  have 

cy 


(x- 


ex 


c±R 


■{y 


c-±R 


)-=R^ 


222  DIFFERENTIAL    CALCULUS. 

whence 

the  equation  of  the  envelope  showing  it  to  be  twofold.  The 
positive  value  of  R  gives  a  circle  with  a  radiu.s  equal  to  <:+K. 
circumscribing  the  system,  and  the  negative  value  for  R 
gives  one  that  is  inscribed  within  it. 

( S  23)  Let  there  be  an  ellipse  in  which  the  axes  vary  in 
length  under  the  condition  that  the  area  of  the  ellipse  shall 
be  constant.  This  condition  will  be  expressed  by  the  equa- 
tion 

To  find  the  envelope  of  this  curve  we  put  its  equation 
under  the  form 

2  2 

£!._l21_  /  X 

A^     B2~^  V^; 

and  differentiating  with  respect  to  A  and  B  only  we  have 

x^      y'^      ^/B 
A^"^B3"  •  .Ta^"^ 
or 

I        x^  I       ^B     y'^ 

~K   •  A^~~"b'  •  ^  •  Bs" 
But  from  equation  (i)  we  obtain 

A~~  B   •  ^A 

whence 

X^  j;3 

whence 

A=^X'\/2'  and  B=jV2' 
Substituting  these  values  in  equation  (i)  we  have 

2xy=AB'-=c^ 
and 

.2 


C 

xy=^ 


2 


ENVELOPES.  223 

which  is  the  equation  of  a  hyperbola  referred  to  its  center 
and  asymptotes.  The  curve  EF  (Fig.  49)  is  then  a  hyper- 
bohi,  and  the  axes  of  the  elHpse  are  its  asymptotes. 

(124)  Let  AB  (Fig.  48)  and  AC  be  the  coordinate  axes, 
and  let  the  line  de  of  a  given  length  move  in  such  a  manner 
that  its  extremities  shall  be  at  all  times  in  the  axis.  What 
is  the  equation  of  the  envelope  described  by  that  line? 

Call  the  length  of  the  line  c,  and  the  distances  A^/'and 
Ke  respectively  b  and  a.  Let  A;;/=jc  and  7nn—y,  then  the 
general  equation  of  the  line  will  be 

we  have  also 

Differentiating  these  equations  with  respect  to  a  and  I?  as 
variables  we  have 

db a  _  b"X 

da      b      a'^y 

or 

y      b'^^x 

'b^~a^ 
Substituting  this  value  in  equation  (i)  we  have 

X      b~x 


a 


whence  we  obtain 
and  similarly 
whence 


ci  —  Vc"x 


b=\  c^ 


y 


from  which 


2       J.      |. 


.^3  _|_j;3  z=:(^'i 

which  is  the  equation  of  the  envelope. 


224 


DIFFERENTIAL    CALCULUS. 


The  first  differential  coefficient  of  this  equation  is 

-1  4 


dy 


y 


from   which   we  learn   that  the   curve    is    tangent    to   both 
coordinates. 

(125)  Suppose  the  line  DC  (Fig.  52)  to  revolve  about  the 
point  D  in  the  axis  of  abscissas, 
varying  in  length  so  that  the 
extremity  C  shall  be  at  all  times 
in  the  axis  of  ordinates,  required 
the  envelope  described  by  the 
line  CE  perpendicular  to  DC  at 
the  point  C  in  the  axis  of  ordi- 
nates. 

Representing  the   distance   AD  by  ^,  and  the  tangent  of 

the  angle  CDB  by   — — ,  its  equation  will  be 


y- 


a 


{x-c) 


in  which  if  we  make  x^^o  we  have 

c 


y- 


a 


lor  the  distance  from  the  origin  at  which  the  line  DC  inter- 
sects the  axis  of  ordinates.  And  since  the  perpendicular 
passes  through  the  same  point,  its  equation  will  be 


c 

■ax-\-~ 
a 


(i) 


If  we  consider  a  in  this  equation  as  an  independent  variable, 
it  will  represent  all  the  perpendiculars  that  can  be  drawn 
under  the  given  condition. 

Differentiating  it  with  respect  to  a  we  have 


X—- 


a' 


ENVELOPES. 


22; 


whence 

and  substituting  this  value  of  a  in  equation  (i)  we  have 


whence 


y^—Acx 


which  shows  the  envelope  to  be  a  parabola  of  which  D  is  the 
focus.  It  also  demonstrates  a  well  known  property  of  the 
parabola,  namely,  if  lines  be  drawn  from  the  focus  perpen- 
dicular to  the  tangent  they  will  intersect  it  on  the  perpen- 
dicular to  the  axis  through  the  vertex. 

(126)  Let  AB  and  EO  (Fig.  53)  be  the  coordinate  axes, 
and  let  CD  be  a  line  revolvins 
between  the  lines  AH  and  BK 
in  such  a  manner  that  its  ex- 
tremities C  and  D  shall  always 
be  in  those  lines,  and  the  pro- 
duct of  the  distances  CA  and  ~" 
DB  from   the   axis   shall    be    a  ^^s-  53. 

constant  quantity.  Required  the  equation  of  the  envelope 
generated. 

Let  OA=OB=;;/,  and  AC  .  BD^^^r^  Then  producing  the 
line  DC  until  it  meets  the  axis  of  abscissas  at  S,  and  mak- 
ing the  tangent  of  BSD=^?,  we  have 

SB:BD::SO:OF 


or 


whence 

and  similarly 


OF  OF 

—  +w:BD:: —  :  OF 
a  a 


BD=OF+^w 


2  26  DIFFERENTIAL     CALCULUS. 

whence 

or 

But  the  equation  of  the  line  CD  is 

in  which  ^  is  the  distance  from  the  origin  to  the  point  where 
the  line  cuts  the  axis  of  ordinates,  that  is,  the  distance  OF. 
Hence 

is  the  equation  which,  when  a  is  variable,  represents  the  line 
CD  in  every  position  it  can  assume  under  the  given  condi- 
tions. 

Differentiating  with  respect  to  a,  we  have 

m^ada 
xda-\- 1~° 

whence 

a=  —  — 


c  X 

m 


which  being  substituted  in  equation  (i)  gives 

ym^—x^Y 


whence 

whence 
whence 


_Ji  C' 

y{in^  —  x'^y^^^ — —x'^-\-C7n 


m 


my(7n'^—x^)^=c{m^—x^) 


7/ry  ^=^m^c''  —c^X" 


ENVELOPES.  227 

or 

Iffy  -j-cx'^  — m-c 
which  is  the  equation  of  an  ellipse  referred  to  its  center  as 
the  origin,  and  whose  semi-axes  are  7/1  and  c. 

(127)  The  equation  of  the  normal  line  to  the  parabola  is 

y-y=-j(.X-x')  (l) 

in  which  x  and  /  are  the  coordinates  of  the  point  in  the 
curve  from  which  the  normal  is  drawn,  and  x  and  y  are  the 
variable  coordinates  of  the  normal  itself. 

If  we  consider  x'  and  y  as  variables,  equation  (i)  will 
represent  the  entire  system  of  normals  which  can  be  drawn 
to  the  parabola.  To  find  the  envelope  of  this  system  we 
find  the  relation  between  x'  and  y  from  the  equation  of  the 
parabola 

y'^-'—ZpX  (2) 

and  substitute  in  equation  (i)  the  value  of  x  ^  which  gives 

y  X     y^ 

whence 

2p'^{y—y')  =  -  2pxv  +y  ^  (3) 

Differentiating  this  equation  with  respect  toy   only  we  have 

—  2/>*  =  —  2px + 3  j/2 
whence 

y=Vi{p^^p^) 
Substituting  this  value  in  equation  (3)  we  have 

whence 

2fy+2{px-p^)(i{px-p^yf=\^{px-p^y^^ 

whence 
whence 


•i 


2  28  DIFFERENTIAL    CALCULUS. 


or 


which  as  we  have  seen  (Art.    115)   is  the  equation  of  the 
evoliite  of  the  parabola. 

■  Hence  all  normal  lines  to  the  parabola  are  tangent  to  the 
evolute. 


SECTION   XV. 


APPLICATION  OF    THE  DIFFERENTIAL    CALCULUS  TO 
THE   DISCUSSION   OF  CURVES. 

THE    CYCLOID. 


( 1 28)  The  cycloid  is  a  curve  described  by  any  point  in 
the  circumference  of  a  circle  as  it  rolls  along  a  straight  line 

If  for  example,  the  circle  EFD  (Fig.  54)  should  roll  along 
the  straight  line  AB, 
the  point  F,  starting 
from  the  point  A, 
would  describe  the 
cycloid  AD'B,  and 
the  distance  from  A 
to  B  where  the  gen- 
erating point  again  meets  the  line  AB  will  be  exactly  equal 
to  the  circumference  of  the  generating  circle. 

If  we  place  the  origin  at  A  we  shall  have 
AG=A^  and  FG=j^ 
The  arc  FE  will  be  equal  to  the  line  AE,  and  HE  will  be 
the  versed  sine  of  the  same  arc.     Making  DE  =  2/'-  we  shall 
have 

FH^=DH  .  YiY.^y{2r-y) 

229 


230  DIFFERENTIAL    CALCULUS. 

hence 


FH=GE  =  arc  FE—x  =  ^2ry—y^ 


whence 


-1  ______ 

x—YQV.  sin.     7—^/2/7—7"  (i) 

which  is  the  equation  of  the  cycloid. 

The  line  AB  is  called  the  base  of  the  cycloid,  and  the  line 
D'E'  perpendicular  to  the  base  at  its  middle  point  is  the 
axis,  and  is  equal  to  2r. 

Since  every  negative  value  for  jj'  gives  an  imaginary  value 

for  X,  the  curve  has  no  point  below  the  base.     If  we  make 

y=^2r  we  have 

-1 
^=ver   sin.      2r=:rr 

and  every  value  for  y  greater  than  2r  gives  an  imaginary 

value  for  x;  hence  the  greatest  value  of  j'  is  the  diameter 

of  the  generating  circle ;  and  for  all  values  of  y  between  2r 

and  zero  there  will  be  a  real  value  for  x. 

( 1 29)  We  will  now  proceed,  with  the  aid  of  the  differen- 
tial calculus,  to  investigate  the  properties  of  this  curve  in 
reference  to  its  tangent,  subtangent,  normal,  subnormal, 
curvature,  evolute,  etc. 

Differentiating  equation  (i)  we  have 

ri(y  luiy—ydy  ydy 

y  27'y — y"     'V  27y — y'^      y27y—y'^  (2) 

Substituting  this  value  of  dx  in  the  general  formula  for  the 
subtangent  (Art.  52)  we  have 

TG-        ^ 


V  2;y  — y'^ 
and  for  the  tangent  (Art.  53) 


TF 


-■\/r 


y' 


2  7'y  — y  ^ 


For  the  subnormal  (Art.  54) 

GE=\/2;-y — J/3 


DISCUSSION    OF    CURVKS.  23 1 

and  for  the  normal  (Art.  55) 

Since  GE  the  subnormai  is  equal  to  y\/2)y  —  j-,  which  is 
equal  to  y^DH.  HE,  the  point  E  of  the  subnormal  for  the 
point  F  of  the  curve,  must  be  at  the  intersection  of  the  ver- 
tical diameter  of  the  corresponding  generating  circle  with 
the  base  ;  and  the  normal  line  ^V'sry— Vde.  EU  must  be 
u  chord  of  thai  circle  joining  these  two  points. 

The  tangent  being  perpendicular  to  the  normal  will  of 
course  be  the  supi)lementary  chord  of  the  same  circle. 
Hence  to  obtain  the  normal  and  tangent  lines  for  any  given 
point  of  the  cycloid,  construct  the  generating  circle  for  the 
diameter  D'E'  erected  at  the  middle  of  the  base,  and 
through  the  given  point  draw  the  line  FH'  parallel  to  the 
base  intersecting  the  circle  at  F'.  Join  this  point  with  the 
extremities  of  the  diameter  D'E',  and  the  line  F'E'  will  be 
parallel  to  the  normal,  and  F'D'  will  be  parallel  to  the  tan- 
gent. Hence  lines  parallel  to  these,  through  the  given  point 
will  be  the  lines  required. 

If  it  is  required  to  draw  a  tangent  parallel  to  a  given  line, 
first  draw  a  chord  from  D'  parallel  to  the  given  line,  and 
through  the  point  where  it  meets  the  circumference  of  the 
circle  draw  a  line  parallel  to  the  base.  The  intersection  of 
this  line  with  the  curve  of  the  cycloid  will  be  the  point  of 
tangency. 

(130)  From  equation  (2)  we  have 


dy      y  2ry— y'^ 

dx~        y         ~  "^     y 


=\/i^-'  (3) 


which  becomes  zero  when  ^  =  2/',  hence  the  tangent  at  the 

extremity  of  the  axis  is  parallel  to  the  base.     If  we  make 

y^=^o  we  have 

dy 


—  =  00 
ax 


232  DIFFERENTIAL     CALCULUS. 

hence    the    tangent    at    the    base    is    perpendicular   to    it. 
Differentiating  equation  (3)  we  have 

2  rdy  rdy 

d "  y  y^^  y'^  rdx 


dx  ^     /  2r  ^  y 

dx 


Vv- 


hence 


d'^  y  r 

dx^  y'^ 


This  second  differential  coefficient  being  essentially  neg- 
ative, shows  that  the  curve  is  everywhere  concave  toward 
the  base. 

(131)  The  formula  for  the  radius  of  curvature  (Art.  105) 

gives  in  this  case, 

3 
2rdx'^  ^A     /2j'dx^\^ 

Tj  — -^z. iz:  2    /'    V 

rdx^  rdx'^  -^ 


y2,  y2 


or 


R  =  2^2/7 

But  we  have  found  (Art.  129)  the  normal  to  be  equal  10 
^/~2ry ;  hence  the  radius  of  curvature  at  any  point  is  equal 
to  twice  the  normal  at  that  point.  Thus  at  A  the  radius  of 
curvature  is  nothing,  while  at  D'  it  is  equal  to  2D'E'=4r. 

( 1 32)  The  equation  of  the  evolute  will  be  found  by  the 
rule  given  in  Art.  114. 

In  the  equations  of  condition  (Art.  105) 

dv  ,         .  [  . 

and 

dx'^  -\-dy^ 

y—h  —— — 7:; (3) 

Substitute  the  values  of  -4^  and  Vs"  J^^^  ^^^^^^  ^'^^"^ 

dx   .        dx'^ 


DISCUSSION    OF    CURVES. 


^2>Z 


the   equation  of  its  cycloid,   and   then,   by  means  of  that 
equation  eliminate  x  and  y. 
Thus 

\  2ry — y'^ 


and 


x—a^- 


y 


-0'-^) 


y — b=^ 


dx^  -\-(ix~  (  — —  I  ) 
^  y       ' 

dx-r 


whence 


or 


y—b'=^2y  and  x—a^^  —  i's/' 2,-y—y% 


y=^—b  and  x^a—2^  —  2rl?—f?^ 
Substituting  these  values  of  x  andj'  in  equation  (i)  (Art. 
128),  we  have 

:ver.  sin." 


a—W—2rl?—b^ 


-1 


■b—  ^  —2rb—b'' 


or 


i?  =  ver.  ^\\\r^—b-\'^  —  2rb—b'^ 


(4) 


which  is  the  equation  of  the  e volute. 

(133)  For  all  values  of  b  that  are  positive  a  is  imaginary, 
hence  no  part  of  the  curve  is  above  the  base  of  the  invo- 
lute. For  all  negative  values  of  b  greater  than  2r,  a  is  also 
imaginary,  hence  if  we  draw  A'B'  (Fig.  55)  parallel  to  the 
base  at  a  distance 
below  it  equal  to 
2/-,  the  evolute 
will  lie  between 
that  line  and  the 
base.  If  we  make 
<^  =  —  2r,  a  be- 
comes equal  to 
the  arc  whose 
versed  sine  is  —b^ 


234  DIFFERENTIAL    CALCULUS. 

that  is  half  the  circumference  of  the  generating  circle. 
Hence  the  point  G  where  the  evolute  meets  the  line  A'B'  is 
in  the  prolonged  axis  of  the  involute.  If  we  make 
(^~o,  a  also  becomes  equal  to  zero,  and  hence  the  evolute 
passes  through  the  origin  at  A,  and  also  the  extremity  of  the 
base  at  B.  For  ver.  sin.~i<?  may  be  zero,  or  it  may  be  a 
whole  circumference. 

If  we  differentiate  equation  (4)  we  have 

_  /v//^  rdb-\-bdb  {2r-{-b)db 

da  —  —    /  —    /  = ,  — 

\  —  2rb—b^     V  —  2rb—b^  V —  2rb—b^ 

or 

db__     V  -2rb-^ 

da  2r-\-b 

showing  that  at  the  points  C  and  B  where  b=^o  the  base  of 
the  involute  is  tangent  to  the  evolute.     Also  since 


^-zrb-b-^  b 


2r-{-b  ^ —  2rb—b^ 

if  we  make  b^  —  2r  we  have 

db 


da 


showing  the  tangent  to  the  evolute  at  G  is  perpendicular  to 

the  line  A'B'. 

db 
Squaring  the  value  of  -y-,  and  differentiating,  we  have 

im  _  b  d^b_  r 

da"  2r+ b  da"  {2r-\-bY 

which  is  essentially  negative,  and  since  every  real  value  of 
b  is  also  negative,  the  curve  is  everywhere  convex  to  the 
base  of  the  cycloid. 

( 1 34)  These  circumstances,  together  with  the  form  of  the 
equation  of  the  evolute,  lead  us  to  suppose  it  to  be  an  equal 
cycloid,  but  for  certainty  we  will  transfer  the  origin  to  G, 
and  the  coordinate  axes  to   EG  and  GF  respectively  par- 


DISCUSSION    OF    CURVES.  27,^ 

allel  to  the  first.     Calling  the  new  coordinates  x'  and  j''  we 

have 

a=^x^-\-m  and  d—y'-\-n 

m  and  n  being  the  coordinates  of  the  new  origin  referred  to 
the  original  axes.     Then 

;/z  =  ver.  sin.'~^2;"  and  ;/  =  —  2r 
whence 

rt;=^'  +  ver.  sin.~^2/-  and  /.»  =  — (2;-— y') 

Substituting  these  values  oi  a  and  b  in  equation  (4)  we  have 
jv'  +  ver.  sin.~i2/'  =  ver.  sin.~ ^ (2;-— j'')  +  y  2r{2r—y')—{2r—y)^ 
but 

ver.  sin.~i2;'— ver,  sin,~ i( 2;-— _)/')  =  ver.  sin.""iy 
hence 

:r'=  — ver.  sin .  ~  ^j-'  +  y  2 ry'  —y  ^ 

which  is  the  equation  of  the  curve  CG,  the  values  of  x'  be- 
ing the  same  as  those  o(  x  in  equation  (i)  (Art.  128),  except 
that  they  are  negative  as  they  should  be,  since  the  values 
of  X  are  reckoned  in  a  contrary  direction  from  those  of  x ; 
and  the  curve  CG  is  equal  to  the  curve  CF,  but  reversed  in 
position  with  reference  to  the  origin. 

Since  the  curve  CG  is  equal  to  FG  (Art.  113)  the  length 
of  the  cycloid  is  equal  to  four  times  the  diameter  of  the 
generating  circle. 

( 8  35)  The  character  of  the  evolute  of  the  cycloid  may  be 
demonstrated  geometrically  thus  : 

Let  us  suppose  two  right  lines  AB  and  A'B'  (Fig.  55)  to 
be  drawn  parallel  to  each  other,  and  two  circles  to  be  des- 
cribed on  the  diameters  DC  and  CE,  each  equal  to  the 
distance  between  the  two  parallel  lines  and  tangent  to  each 
other  at  the  point  C.  If  now  we  suppose  each  circle  to  roll 
along  the  line  on  which  it  stands,  at  the  same  rate,  so  that 
they  are  at  all  times  tangent  to  each  other,  then  the  point 
C  of  the  upper  circle  will  describe  the  first  half  of  a  cycloid 
CPF,  while  the  same  point  C  of  the  lower  circle  will  des- 
cribe the  last  half  of  an  equal   cycloid  CP'G. 


236  DIFFERENTIAL    CALCULUS. 

Suppose  the  two  circles  to  have  arrived  at  the  point  C  in 
the  line  AB,  and  that  P  is  a  point  in  the  upper  curve.  The 
diameter  DC  of  the  upper  circle  will  have  assumed  the 
position  PO,  and  the  diameter  CE  of  the  lower  circle  will 
have  assumed  the  position  O'P'  parallel  to  it;  and  P' wilt 
be  the  generating  point  of  the  lower  cycloid. 

Draw  the  chord  PC'  and  it  will  be  normal  to  the  upper 
cycloid  (Art.  129).  Draw  also  the  chord  C'P',  and  it  will 
be  tangent  to  the  lower  cycloid  at  the  point  P'  (Art.  129). 
Now  since  PO  and  O'P'  are  parallel,  these  two  chords  and 
the  corresponding  arcs  are  equal,  and  hence  the  angles 
PC'D'  and  P'C'E'  are  equal;  and  since  D'E'  is  a  straight 
line  P'C'P  is  a  straight  line  also,  normal  to  the  upper  curve 
and  tangent  to  the  lower  one.  Hence  the  lower  cycloid  is 
the  evolute  of  the  upper  one. 

(S36)  The  equation  of  the  evolute  may  also  be  obtained 
by  considering  it  as  the  envelope  of  the  normals  drawn  to 
the  curve. 

The  general  equation  of  the  normal  to  the  cycloid  is 

V 

y-y  =  -^/    ^      rS^-^')  (i) 

V  2; J  — y  '^ 
in  which  x'  and  y'  are  the  coordinates  of  that  point  of  the 
cycloid  to  which  the  normal  is  drawn ;  and  x  andjj^  the  gen- 
eral coordinates  of  the  normal  line.  If  we  make  x  and  y' 
variables,  still  retaining  their  relative  values,  as  in  the  equa- 
tion of  the  cycloid,  the  equation  (i)  will  represent  the  whole 
system  of  normals  that  can  be  drawn  to  the  curve.  If  now  we 
eliminate  one  and  make  the  differentials  of  the  equation 
with  respect  to  the  other  equal  to  zero,  then  (Art.  120)  by 
eliminating  that  we  shall  have  an  equation  which  will  be 
that  of  the  envelope  of  the  normals,  and  also  the  evolute  of 
the  cycloid. 

Substituting  for  x'  in  equation  (i)  its  value  taken  from 
the  equation  of  the  curve  (Art.  128)  we  have 


DISCUSSION    OF    CURVES.  237 


y—y= —  ,        -(x—ver.  sin -y+V  2;-/ — /") 

V  2ry  — y 

or 

^_— y^r+y ver.  sin.~^/     , 


y-y  - — w  ■  .     77 — -y 

V  2ry  —  y 


whence 


y\  2rv'  —  j''~— j''ver.  sin.~y +-^y  =0  (2) 

Differentiating  tiiis  equation  with  respect  to  /  we  have 

V  2ry  — y  ^  V  2ry  — y  ^ 

Substitutingin  this  equation  for  ver.  sin.~ij''its  value  taken 
from  the  equation  of  the  cycloid,  and  multiolying  by 
y  27y' — y''^  ^  we  have 

j(r  — y )  -  [x  +  v  2rj;'_y2)  \/  2/y-y2 — r'  ^'^ +^v  v  ^7737^ =o 

or 

j'(r— y )  =^V 2/7  —y  ~  +2ry  — y  2  +y /'— .tV 2ry'  —y  ^ 
or 

x^— y)=-(-^-'^ov2/'y-y~+3^y-y^ 

but 

y  —  y  /  — > — r- 

X — x''=^  — r-V  2ry  — v'^ 

y 

hence 


y{r—/)=      ^,     {2ry—y-)+^ry—y^ 

clearing  of  fractions  and  multiplying  we  have 

ryy  —yy  ''=^2ryy  — 2ry  " —yy  " -f-y    -rs^y  " — y 
whence 

ryy  = — ry  "^  or  y^=^^y 
Substituting  this  value  of  j''  in  equation  (2)  we  have 

y's/  —  2)y — _y"  +_y  ver.  sin.~'^  —y — xy^=^o 
or 


.%'=ver.  sin.-^ — >'  + V — iry — y 


238 


DIFFERENTIAL    CALCULUS. 


which  is  the  equation  of  the  envelope  of  the  normals,  and 
also  of  the  evolute  of  the  cycloid,  as  in  Art.  132;  for  sub- 
stituting the  variables  a  and  b  for  x  and  7,  the  equations  are 
identical. 


THE    LOGARITHMIC    CURVE. 


( 1 37)  The  logarithmic 
curve  is  one  in  which  one 
of  the  coordinates  is  the 
logarithm  of  the  other. 

Its  equation  is 

^=Log.  _y 
If  we  represent  the  base  of 
the  system  by  a  the  equa-  □ 
tion  may  be  written 


y^d^ 


-5-^-^-2.-}   API 
Fig.  56. 


2-  T    ^A   5    B 


The  curve  may  be  constructed  by  laying  off  on  AB  (Fig. 
56)  the  axis  of  logarithms,  the  numbers  i,  2,  3,  4,  etc.,  on 
both  sides  of  the  origin,  and  laying  off  on  the  corresponding 
ordinates,  or  on  AC  the  axis  of  numbers,  the  corresponding 
powers  of  a. 

When  x=o  then  j'^i,  whatever  may  be  the  value  of  a, 
and  hence  all  logarithmic  curves  will  intersect  the  axis  of 
numbers  at  a  distance  from  the  origin  equal  to  i. 

If  a  is  greater  than  i,  and  x  positive,  y  will  increase  as  x 
Increases,  and  there  will  be  a  real  value  of  y  for  every  value 
of  X  as  in  the  curve  DE. 

If  X  is  negative,  then  the  value  of  y  is  fractional,  and 
decreases  as  x  increases  negatively,  but  y  will  not  become 
zero  until  ^=—00. 

If  y  is  negative,  there  is  no  corresponding  value  of  x,  and 
hence  the  curve  can  neve/r  pass  below  the  axis  AB. 

If  a  is  less  than  i,  then  y  will  diminish  as  x  increases 


DISCUSSION    OF    CURVES.  239 

positively,  and  becomes  zero  when  x^=  oo  ;  but  y  increases 
for  negative  values  of  x,  and  the  curve  has  a  position  the 
reverse  of  the  first  as  DD"  in  the  figure. 

( 1 38)  If  we  differentiate  the  equation 

we  have 

dy  T   I 

t/x  ni       "  m 

and 

d^  y  T  I 

dv 
If  we  make  y^=^o  we  have  -i— =o,  hence  the  tangent  for 

that  value  oi y  is  the  axis  of  abscissas  ;  and  since  j'=o  gives 

x^=- —  00  the  axis  of  abscissas  is  an  asymptote  to  the  curve 

dy 
(Art.  88).     But  since  y^  oo  gives  x=^  oo,  and  also  ~i-=  oo, 

the  curve  has  no  tangent  parallel  to  the  axis  of  ordinates 
except  at  an  infinite  distance.  The  sign  of  the  second  dif- 
ferential coefftcient  shows  that  the  curve  is  at  all  times  con- 
vex toward  the  axis  of  abscissas. 

dx 
The  subtangent  PT=-t~j^  =  M;  hence  the  subtangent  is 

constant  and  equal  to  the  modulus  of  the  system  of  logar- 
ithms, to  which  the  curve  belongs. 

In  the  Naperian  system  the  modulus  is  i,  and  in  this  case 
PT  and  DA  are  equal. 

( 1 39)  We  will  now  investigate  the  curve  whose  equation  is 

y^^x  log.  X 
Every  value  for  x  gives  a  single  value  for  y. 

If  X  is  less  than  i  the  value  oi  y  is  negative.  (i) 

If  X  is  greater  than  i,  j  is  positive.  (2) 

If  jt:=o,  or  ^=:i,  y^^o.  (3) 

If  X  is  negative,  y  is  imaginary.  (4) 


240 


DIFFERENTIAL    CALCULUS. 


If  we  differentiate  the  equation  we  have 

dy 

X 


dx'^ 


dy 


Making  -7— =0  we  have 


dx 


loR.  Jt:  =  —  I  or  x=^e~^  =— ^ — — 3— 

°  e       2.7182 


(5) 

(6) 
(7) 


which  corresponds  to  a  minimum  as  shown  by  the  positive 
d'^  y 


value  of 


ax" 


When  x^o, 


d^ 

dx 


00, 


(8) 

(9) 
(.0) 


dy 
When  Jt:  =  I ,  -;— =  I . 

Since _>"  is  negative  between  x^=o  and  x=^i  and  then  pos- 

.       d  "  V    .         . 

itive,  while -p|-  is   always    positive,  the  curve  is  concave 
toward  the   axis  of  abscissas  between  x=-o  and  x=-i^  and 


afterwards  convex. 

Hence  the  curve  begins  at  the 
origin  (Fig.  57)  and  intersects  the 
axis  of  abscissas  at  D,  making 
AD  =  i  (3).  The  tangent  to  the 
curve  at  D  makes  an  angle  of  45° 
with  the  axis  of  abscissas  (10), 
while  at  A  the  axis  of  ordinates  is 
tangent  (9).  At  the  point  E,  whose 
I 


(11) 


l"'ig-  57- 


abscissa  is 


,  the  tangent  to  the  curve  is  parallel  to 


2.7182 

the  axis  of  abscissas  (7),  and  the  ordinate  is  at  a  minimum 
(or  negative  maximum)  (S).  Between  A  and  D  the  curve  is 
below  the  axis  of  abscissas  (i),  and  concave  to  it  (n);  and 


DISCUSSION    OF    CURVES. 


241 


beyond  D  tlie  carve  lies  entirely  above  the  axis  of  abscissas 


an 


d  is  convex  to  it.     Since  x=^ 77—  erives  y  = — x\  we  have 

2.7182  °        -^ 


FE=AF 
(140)  We  will  next  take  the  equation 

__  ~X         I 


Every  value  of  x  gives  a  real  positive  value  for  y,  and 
hence  there  can  be  no  negative  value  of  7.  (i) 

If  .v=o,  y=o,  and  hence  the  curve  passes  through  the 
origia.  (2) 

If  .r  is  negative  we  have  y=^e^ ,  in  which  if  x^^o,  y  will 
become  infinite.  (3) 

So  that  .v=o  gives  two  values  for  y,  according  as  x  ap- 
proaches zero  from  the  positive  or  negative  side.  (4) 

If  X  be  negative  and  increase  in  value,  that  of  y  will 
approach  more  nearly  to  i,  which  it  will  reach  when 
X——QO.  (5) 

If  X  be  positive,  and  increasing  the  value  of  j'  approaches 
more  nearly  to  i,  which  it  reaches  when  x^=  00.  (6) 

Hence  the  curve  will  be  as  in 
Fig.  58,  in  which  AB  and  AC  are 
the  axes  of  coordinates,  and  DE  a 
line  parallel  to  AB  at  a  distance 
from  it  equal  to  i.  It  will  pass 
through  the  origin  A  (2),  extend 
indefinitely  in  a  positive  and  nega- 
tive direction,  and  the  line  DE  will 
be  an  asymptote  to  both  branches  (5),  (6).  The  axis  of 
ordinates  will  also  be  an  asymptote  in  the  positive  direction 
(3)  (Art.  88).  As  the  branch  DC  of  the  curve  extends  to  an 
infinite  distance  in  both  directions,  it  has  no  connection  with 


242  DIFFERENTIAL    CALCULUS. 

the  branch  AE,  which  commences  at  the  origin  and  is  infi- 
nite at  the  other  extremity.  There  are,  in  fact,  two  curves, 
one  answering  to  the  positive,  and  the  other  to  the  negative 

value  of  ^. 

_j_ 

If  we  differentiate  the  equation  j|;=<?    ^  we  have 

1 


and 


dy 

e    ^ 

dx  ~ 

"  x^ 

d"^  y     e 

1 

■2X) 

dx^ 

x^ 

Since 


we  shall  have 


x^'    ~        1 
x^e'^ 

dy  _ 

dx 


when  either  x^=-o  or  :r=  00,  hence  the  axis  of  abscissas  is 
tangent  at  the  origin,  and  parallel  to  the  tangent  at  an  infi- 
nite distance  in  either  direction;  in  which  casej^  =  i  (5)  (6). 

For  all  negative  values  of  x^  ,  ^  is  positive,  and  hence 
the  branch  DC  is  convex  to  the  axis  of  abscissas.  For  all 
positive  values  of  x  less  than  f ,  ,  ,0  is  also  a  positive,  show- 
ing that  between  A  and  H  the  curve  is  convex  to  the  axis 
of  abscissas,  while  at  the  point  H,  where  ^==1,  the  value  of 

^^  y 
,  2   changes  from  positive  to  negative  passing  through  zero, 

showing  that  at  P  the  curve  ceases  to  be  convex,  and  be- 
comes concave  toward  the  axis  of  abscissas.  This  is  called 
an  inflexion. 


SECTION    XVI. 


SINGULAR  POINTS. 

(841)  Singular  points  of  a  curve  are  those  at  which  there 
exists  some  remarkable  property  not  common  to  other  points 
of  it.  Such,  for  example,  as  the  maximum  or  minimum 
value  of  the  ordinates  or  abscissas,  points  of  inflexion,  con- 
jugate points,  cusps,  etc. 

In  many  cases  these  points  are  easily  discovered  by  the 
aid  of  the  differential  calculus,  as  will  be  seen  by  the  fol- 
lowing examples. 

MAXIMA    AND    MINIMA. 

(142)  If  we  differentiate  the  equation 
7=3  +  2(a:-4)^ 
we  shall  have 

d^  y 

d^y 
^^=48(^-4) 

d^y 

dx^       ^ 

Here  we  find  that  .t=4  will  reduce  the  first  differential 

243 


^^_^r=24(-v-4)= 


244 


DIFFERENTIAL    CALCULUS. 


coefficient  to  zero,  showing  that  the  tangent  to  the  curve  is 
parallel  to  the  axis  of  abscissas  (Art.  ^6),  and  hence  the 
value  of  the  ordinate  may  be  a  maximum  or  minimum.  But 
since  the  second  differential  coefficient  is  always  positive 
except  when  it  is  zero,  the  first  must  be  an 
increasing  function,  and  hence  at  zero  must 
be  passing  from  negative  to  positive,  and  the 
value  of  y  must  be  changing  from  a  diminish- 
ing to  an  increasing  one.  So  that  there  is  a 
minimum  when  x=^4,  as  shown  in  Fig.  59, 
We  infer  the  same  thing  from  the  sign  of  the  fourth  differen- 
tial coefficient  (Art.  29). 
If  we  take  the  equation 

y=^2  —  2{x — 2)* 


A 


Fi£ 


59- 


-we  shall  have 


d^  y 

d^y_ 
dx^  ' 


2)3 

— 48(x— 2) 
-48 


Since  jr  =  2  reduces  the  first  differential  coefficient  to  zero 
the  tangent  at  that  point  is  parallel  to  the  axis 
of  abscissas;  and  since  the  fourth  differential 
coefficient  (the  first  that  has  real  value  for 
x=2)  is  negative,  the  value  oi  y  at  that  point 
must  be  a  maximum  as  in  the  figure.     Since 

d^ y  , 

—r-2  IS  at  all  times  negative,  except  when  ^=2,  the  curve 

will  be  concave  toward  the  axis  of  abscissas  for  all  positive 
values  of  J  (Art.  91). 


r 

Fig.  60. 


SINGULAR    POINTS.  245 


POINTS    OF    INFLEXION. 


(143)  A  point  of  inflexion  is  one  in  which  the  radius  of 
curvature  changes  from  one  side  to  the  other  of  the  curve 
so  that  it  will  be  convex  on  one  side  of  the  point  of  inflex- 
ion, and  concave  on  the  other  towards  any  line  not  passing 
through  the  point  itself,  and  this  will,  of  course,  be  true  for 
the  axis  of  abscissas,  and  hence  at  such  a  point  the  second 
differential  coefflcient  will  change  its  sign.  If  the  point  of 
inflexion  should  be  in  the  axis  of  abscissas,  both  parts  of 
the  curve  would  be  convex  or  concave  to  it,  but  the  second 
differential  coefficient  will  still  change  its  sign  (Art.  93). 
Now  in  order  that  the  sign  should  be  changed,  the  function 
must  pass  through  zero  or  infinity,  and  hence  the  equations 

d  ^   ]'  <-/  ~   1' 


Tr=o  and 


ax-  ax" 

will  give  all  the   points  of  inflexion  in  any  curve  in  which 
there  may  be  such  points. 

( 1 44)  Let  us  now  take  the  equation 
y—i-{-T,{x—2Y 
whence 

and 

d  "^  y 

dx^ 
In  this  case  every  value  of  y  gives  one  for  x,  and  vice 
versa^  hence   the  curve  has   no  limit.       When    a:=2,  then 

-TT=o  and  v=:i,  so  that  if  we  make  AC  =  2 


and  CD  =  i  (Fig.  61),  the  tangent  at  Dwill 

be  parallel  to  the  axis  AB.     But  ^=2  re-      ~~^        C  3 

d  "  V 
duces  -i-j-    to    zero,    also    indicating  that  ^ig-  6^- 


^4*5  DIFFERENTIAL    CALCULUS. 

there  7nay  be  a  point  of  inflexion,  hence  we   resort  to  the 

d^  y 
value  of  —j-j  which  is  a  positive  constant.     From  this  we 

a  "  y  , 
learn  that  at  zero  ~J~T  ^s  an  increasing  function,  hence  it 

must  pass  from  negative  to  positive,  showing  that  at  the 

.        dv 

same  point  yy  passes  from  a  decreasing  to  an  increasing 

function,  and  hence  does  not  change  its  sign,  but  remains 
positive  both  before  and  after  the  zero  point;  and  this  shows 
that  the  value  of  y  is  an  increasing  function  both  before  and 
after  the  same  point.  There  is,  therefore,  no  maximum  nor 
minimum  for  it  at  that  point. 

d'^  y  .       . 

Since  ,  2  changes  its  sign  at  :r=2  from  negative  to  pos- 
itive, the  curve  will  be  concave  toward  the  axis  of  abscissas 
when  ^<2,  and  convex  when  ^>2,  so  that  at  the  point  D 
Avhere  x^2  the  curvature  changes  its  direction  and  there  is 
an  inflexion. 

(145)  If  we  take  the  same  equation,  and  make  the  last 
term  negative,  we  shall  have 

dy  \  VD 


£^  =  -i8Gt-2)  Vc       B 

"•^  Fig.  62. 

The  point  D   where  jc  =  2  will   still  be  the  point  of  in- 

dy 
flexion,  but   since    -j-    is    negative    for    all    values    of    x 

except  :r  =  2,  the  curve  will  approach  the  axis  of  abscissas 

.         d  '■^  y 
for   all   positive   values  of  y,  except  j^=i,   and  since     ,  g 

is  positive  for  .r<2,  and  negative  for  .v>2,  the  curve  will  be 
convex  toward  AB  between  A  and  C,  and  concave  afterwards, 
as  in  Fig.  62. 


SINGULAR    POINTS.  247 

The  first  differential  coefficient  being  zero  when  x^2,  it 
follows  that  the  tangent  at  that  point  will  be  parallel  to  the 
axis  of  abscissas  (Art.  2,6),  and  hence  the  curve  will  pass 
from  one  side  of  the  tangent  to  the  other  at  the  point  of  tan- 
gency,  and  will  be  convex  to  the  tangent  on  both  sides  of  it. 

(146)   If  we  take  the  equation 

we  have 

dy 6 

5(^—2)5 


d'^  V 


T2 


25(;c— 2)* 
If  we  make  x^=-2  we  have 

dx  d  ~  y 

•^  '   dx  '      dx- 

and  hence  at  the  point  D  where  x  =  2  the  tangent  will  be 

perpendicular  to  the  axis  of  abscissas  (Fig.  6;^), 

dy  .  .  . 

and  since  -j~  is  positive  for  other  values  of  x, 

the  curve  will  leave  tiie  axis  of  abscissas,  for  all  ~\        c       B 

positive  values  oi  y  as  x  increases.     And  since       Fig-  63- 

d "  V  .  .         ^  .  .  .        . 

j^   changes  its  sign  from  positive  to  negative  in   passing 

through  infinity  where  .r  =  2,  the  curve  will  be  convex  toward 
the  axis  of  abscissas  for  .t<2,  and  concave  for  .v>2,  and  at 
x=2  there  will  be  an  inflexion. 

(147)   If  in  the  same  equation  we  make  the  last  term 
negative  we  have  3 

l'  =  2  —  2\X  —  2)** 

and 

dy  6 


dx  ,  xf 

d"  y  _         12 
dx''^  (  N^ 


248 


DIFFERENTIAL    CALCULUS. 


B 


and  the  conditions  will  be  changed  so  that  the  curve  will  be 
reversed.     It  will  now  approach  the  axis  of  v, 

abscissas,  and  the  second  differential  coeffi-  \  p 

cient  will  change   its   sign  from  negative  to 
positive    in    passing   through  infinity  where 
x=^2\   the  curve  will  be  concave   for  .t<2,  A 
and  convex  for  x'>2. 

The  point  D   (Fig.  64)  will   still  be  a  point  of  inflexion, 
and  the  tangent  will  be  perpendicular  to  AB. 

( 1 48)  If  we  take  the  equation 
y  =  {x — 2)^ 
we  have 


Fig.  64. 


dy 


:=3('^— 2)' 


and 


dx 


^^=6(^-2) 


which  all  reduce  to  zero  when  x^2. 

This  shows  that  the  curve  meets  the  axis  of  abscissas  at 
the  point  where  x=^2,  and  that  this  axis  is       1 
tangent  to  it  there.     And   since  the  second 
differential  coefficient   will    have    the    same 
sign  as  y  (both  being  the   same  as  that  of 
x—2),  it  will  change,  from  negative  to  posi-  Fig.  65. 

tive  at  the  point  where  x=^2y  showing  an  inflexion  there,  and 
that  the  curve  is  convex  to  the  axis  of  abscissas  on  both 
sides  of  it. 


A 


B 


CUSPS. 


( 1 48)  A  cusp  is  a  curve  consisting  of  two  branches  start- 
ing from  a  common  point  in  the  same  direction,  and  imme- 
diately diverging  from  a  common  tangent.  They  are  of  two 
kinds,  namely :    Those  in  which  the  branches  are  on  differ- 


SINGULAR    POINTS. 


149 


ent  sides  of  the  tangent,  which  are  cusps  of  the  first  order ; 
and  those  in  which  the  branches  are  both  on  the  same  side 
of  the  tangent,  which  are  cusps  of  the  second  order. 

The  following  are  examples  of  the  first  order. 

Let 


then 


and 


dy  -1 

^  =  2(.r-,)   ^ 

d"y  2 

dx'^ 


\x- 


If  we  make  :r=i  we  have 

dx^ 
dx 


y- 


=  00,   and 
dv 


dx'^ 


■^ —  CO 


For  every  value  of  .r<i,  -^  is  negative,  and  positive  for 

every  value  greater.     The  curve,  therefore,  approaches  the 

axis  of  abscissas,  in  the  first  case,  and  recedes  from  it  in  the 

second  (j' being  positive),  which  indicates  a  minimum,  while 

the   tangent   at  that  point  is  perpendicular  to 

d^  V  . 
the  axis  of  abscissas ;  and  since  "T~|"  is  always 

negative  the  curve  is  concave  toward  the  same 
axis. 

Every  value  of  y  less  than  i  gives  an  imaginary  value  for 
x^  while  every  value  greater  than  i  gives  two  values  for  x^ 
one  less  and  one  as  much  greater  than  i.  Hence  the  curve 
has  two  equal  branches  commencing  at  D  (Fig.  (}())^  where 
they  have  a  common  tangent. 

(150)  If  we  make  the  last  term  negative,   the  signs  of 

dy  d  ~  1'     . 

-7-  and  '"Ti;"  ^^'lil  be  reversed,  and  the  first  will  chanii;e  from 

dx  dx-  ^ 


A       C 

Fie.  66. 


250  DIFFERENTIAL    CALCULUS. 

positive  to  negative  SiS  x  passes  from.T<i   to 

x>i ;  while  at  x=i  the  tangent  is  still  perpen-  aO 

dicular  to  the  axis  of  abscissas.     Any  value  of  y  ^_  _^l 

greater  than  i  will  give  an  imaginary  value  for  Ai  C  P 
X,  while   every   value  less  than    i   will  give  two  ^^'  ^' 

real  values  for  x  equally  distant  from  the  point  C  where 

x=i  (Fig.  67).  The  sign  of  — r-^  being  now  always  posi- 
tive (except  at  x=^i)  shows  that  both  branches  of  the  curve 
are  convex  toward  the  axis  of  abscissas.  These  are  then 
cusps  of  the  first  order. 

(151)  If  we  differentiate  the  equation 

}'  =  2±{x—l)'^  (l) 


ax  ^         ' 

d^  y  ,         x-1 

We  see  from  equation  (i)  that  when  .:^=i,  )'  =  2,  and  when 
x<,\^y  is  imaginary,  while  when  ..y>  i,jv has  two  values,  one 
greater  than  2  and  the  other  as  much  less  ;  so  that  if  DC  (Fig. 
68)  be  drawn  perpendicular  to  AB,  making  DC^2,  the  curve 
will  commence  at  D  and  be  symmetrical  about  the  line  DE, 

and  since  -^  =0  for  the  point  D,  the  line  DE  will  be  tangent 
to  both  branches.     Since  for  every  other  value  of 

x,~T~  has   one   negative  and  one  equal  positive 


value,  one  branch  of  the  curve  will  approach  the  A      C      B~ 

axis  of  abscissas,  and  the  other  recede  from  it  at      ^"^s-  68. 

i/  ^  y 
an  equal   rate.       And  since  for   every   value  of  ..r  >  i ,    ,  g 

has  two  equal  values  with  contrary  signs,  the  positive  cor- 
responding with  the  greatest  value  of  j,  we  infer  that  the 


SINGULAR    POINTS. 


251 


upper  branch  of  the  curve  is  convex,  and  the  lower  branch 
concave,  to  the  axis  of  abscissas,  and  that  the  curve  is  a 
cusp  of  the  first  order. 

( I  52)  If  we  change  the  sign  of  the  last  term  and  make 


the  equation 

3 

y  =  2±(i—x)^ 

we  have 

l=^t(:-.)^ 

«'^v        3 

and  the  curve  will  be  similar,  but  reversed  in  position  as  in 
Fig.  69. 

If  Jk:>i,  J  will  be  imaginary. 

If  :r=o,  j'=3  and  /  =  i. 

d "  V  . 


If  j=o,  ^=1  — .y/^-     Since     ,  3  is  both  pos-  A        ;       -^ 

itive  and  negative  when  xKi  there  is  no  maxi-        Fis-  69. 
imum  nor  minimum  value  for  y. 

( I  53)  The  curve  represented  by  the  equation 

contains  a  cusp  of  the  second  order,  as  well  as  some  other 
singular  properties. 

Solving  this  equation  we  have 


and  by  differentiation  we  have 


(i) 


dy  .    3 

-7— =  2A-±-:i.r^ 
ax 


and 


d  -  r 


-X- 


dx-      "  '    4 
From  equation  (i)  we  find  that  the  curve  passes  through 


252  DIFFERENTIAL     CALCULUS. 

the  origin,  and  does  not  extend  to  the 

negative  side  of  the  axis  of  ordinates. 

Every  positive  value  for  x<.i  gives  two 

real  positive  values  for  y,  while  x=j 

gives  one  positive  value  for  y  and  one  ^, 

equal   to  zero.     Hence   the  curve  has      [ 

two    branches,    both     of    which    pass 

through  the  origin,  and  one  intersects  the  axis  of  abscissas 

at  a  distance  from  the  origin  equal  to  i. 

t/y 
If  we  make  ~y-=^o  we    have  x=o    and   x=^^^.     Hence 

there  are  two  points  in  which  the  tangent  to  the  curve  is 
parallel  to  the  axis  of  abscissas  ;  at  the  origin  where  the 
axis  itself  is  tangent  and  at  the  point  D  (Fig.  70)  whose 
abscissa  is  ^^^^^-f  ;    and   as  the  value  of  x  at  this   point 

derived  from  the  equation  -7-  =0  corresponds  to  the  minus 

sign  in  equation  (i),  the  point  of  tangency  is  on  the  lower 
branch  of  the  curve. 

15    i- 

The  second  differential  coefficient  has  two  values  2+ — x^ 

4 

15    i_ 
and  2— — :r^,of  which  the  first  belongs  to  the  upper  branch 

of  the  curve  and  is  always  positive,  while  the  second  is  pos- 

15    L  . 
itive  so  long  as  — x^  is  less  than  2 ;  that  is  so  long  as  x  is 

d-'y 
less  than  2%%,  which  makes    ,  2  — o.     After  that  it  becomes 

negative ;  showing  that  the  lower  branch  of  the  curve  is 
convex  to  the  axis  of  abscissas,  as  far  as  the  point  whose 
abscissa  is  AE^^-g^^,  and  at  this  point  there  is  an  inflexion, 
the  curve  becoming  concave  to  the  axis  of  abscissas  as  long 
as  y  is  positive  and  convex  afterward.  Hence  at  the  origin 
there  is  a  cusp  of  the  second  species. 


SINGULAR    POINTS. 


OO 


CONJUGATE    POINTS. 

(154)  Conjugate  points  are  those  single  points  which  are 
isolated  from  the  curve,  but  will  satisfy  the  equation. 
If  we  differentiate  the  equation 


y=± 


V 


x~{x—l)) 


(i) 


we  have 


dx~     2V^ 


2b 


d 


y 


dx^ 


=  ±- 


Zx  —  ^b 


AfCl^ix  —  bj^ 

If  we  make  jc  =  o  in  ec^uation  (i)  we  have  7=0,  but  any 
other  value  of  x  less  than  b  will  make  y  imaginary.  Hence 
while  the  origin  will  satisfy  the  equation,  that  point  is  iso- 
lated, having  no  connection  with  the  curve.  We  also  see 
that  X- 


■o  will  give 


4'^^     -b 
dx  ^Z  —  ab 


which  is  imaginary  as  it  should  be,  since  at  that  point  the 
curve  can  have  no  tangent. 
If  we  make  x'—b^  we  have 

dx 

showing  that  the  tangent  at  that  point  is  perpendicular  to 

the  axis  of  abscissas,  while  the  value  of 

y  is  zero.     As  every  positive  value  of 

xy-b  gives  two  equal  values  for  y  with    . 

opposite  signs,  the  curve  is  symmetrical 

about  the  axis  of  abscissas,  and  as  the 

dy 
value  of  —r  has  the  same  si^rn  as  y,  the      * 

dx  ^  -"  Fig.  71. 

curve  departs  from  that  axis  in  both  directions.     If  we  make 
x  negative  the  value  of  y  becomes  imaginary  ;  showing  that 


254  DIFFERENTIAL    CALCULUS. 

the  curve  does  not  extend  to  the  negative  side  of  the  axis  of 
ordmates. 

If  we  make     ,  o  =o,  we  have 

4b 
x^ — 
3 

showing  that  at  the  points  C  and  C  (Fig.  71)  v/hich  lie  in 
the  ordinate  drawn  through  D  at  a  distance  from  the  origin 
equal  to  f<^,  the  curve  has  an  inflexion  in  each  branch,  since 
for  that  value  of  x  we  have 


4/; 


/  ^ 


V    3^ 


Ah 

If  we  make  x<.-^—,  the  second  differential  coefficient  will 
3 

4b 
have  a  sign  contrary  to  that  of  _y.     If:r>— ^    the  signs  will 

be  the  same.     Hence  between  H  and  D  the  curve  is  concave 

toward  the  axis  of  abscissas,  and  convex  beyond  D,  which 

also  shows  an  inflexion. 

dy 
If  we  make  ~t~^o  we  have  3.r==2^,  or 

2b 

3 

This  value  being  substituted  for  x  in  equation  (i)  gives 

an  imagin*ary  value  for  y,  showing  that  there  is  no  point  in 
the  curve  where  the  tangent  is  parallel  to  the  axis  of  abscissas. 


MULTIPLE    POINTS. 

(155)    A  multiple   point  is   one  in  which   two  or  more 

branches  of  a  curve  intersect  each  other.     At  such  a  point 

the  curve  will   always  have  as  many  tangents  as  there  are 

dy 
branches,  and  hence  -rr  must  have  the  same  number  of  val- 


fx 


ues  for  that  point. 


SINGULAR    POINTS. 


255 


Let  us  take  the  equation 

y=/^±{x—a)\/^^'7  wheiQ  a>c  (i) 

then  by  differentiating  we  have 

i/y  X — a 

Yox  x^=-a  and  ^=r  in  equation  (i)  we  havej^=/>'y  hence 
H  and  H'  (Fig.  72)  corresponding  to 
these  values  of  x  and  j'  are  points  in 
the  curve.  For  all  values  of  x<ic 
that  of  y  is  imaginary ;  hence  there 
is  no  part  of  the  curve  between  H  and  a 
the  axis  of  ordinates.  For  every 
value  of  x>^,  except  x'=a^  y  has  two 
values,  one  greater,  and  the  other  as  much  less  than  /; 
Hence  the  curve  is  symmetrical  about  FIH'.     For  x--c  we 


have 


dx 


00,  hence  the  tangent  at   H  is  perpendicular  to 

dv 


the  axis  of  abscissas.      For  x'=-a  we  have  two  values  of 


dx 


equal  to  each  other  with  contrary  signs,  namely,  ^Ic^^c  and 
—  ^ X  —  c.  Hence  at  H'  there  are  two  tangents  making  sup- 
plementary angles  with  the  axis  of  abscissas,  so  that  the  two 
branches  of  the  curve  cross  each  other  at  that  point  in  direc- 


tions symmetrical  with  HH', 


dv 

If  we  make  — 7^  =  0  we  have 
dx 


a-\-2c 


which  shows  that  at  the  point  corresponding  with  the  ordi- 
nate at  E  where  AE  equals  one-third  of  (2AC  +  AB),  the  tan- 
gent is  parallel  to  the  axis  of  abscissas. 

( 1 56)  We  will  close  the  discussion  of  algebraic  curves 
with  that  of  the  equation 

ay"  —x'^"  -\-{b  —  c)x^  -\-bcx^=^o 
Solving  this  equation  with  reference  to  y  we  have 


256 


DIFFERENTIAL     CALCULUS. 


y=±\/' 


'x[x — b){x-\-c) 


and 


fi) 


dx 


^X^  —  2x{b  —  c) — ^r 


^^e 


^V  ax{x—b)(x-\-c) 

If  in  equation  (i)  we  make  x=o,x=^c^,oi'  x=^—c,  we  have 
in  every  case^y^o.  Hence  there  are  three  points,  H,  A  and 
H'  (Fig.  73),  where  the  curves  meet  the  axis  of  abscissas. 

Every  negative  value  of  jr><:  gives  an  imaginary  value 
for  _)',  hence  the  curve  has  no  point  on  the  negative  side  of 
H,  since  AH=^.  Every  negative  value  of  x<.^  will  give 
two  equal  values  for  y  with  opposite  signs ;  hence  from  H 
to  A  the  curve  is  symmetrical  about  the  axis  of  abscissas. 
Every  positive  value  for  x<il?  gives  an 
imaginary  value  for  yy  hence  no  part 
of  the  curve  lies  between  A  and  H'. 
Every  positive  value  for  .T>b  gives 
two  equal  values  for  y  with  contrary 
signs.  Hence  on  the  positive  side  of 
H'  the  curve  is  symmetrical  about 
the  axis  of  abscissas,  and  the  entire  curve  consists  of  two 
parts  having  no  connection  with  each  other  by  a  common 
point. 

Each  of  the  values  of  ^  that  reduce  j^  to  zero  also  reduce 

dy        .        .        ^ 

-J-  to  infinity ;  hence  at  the  points  H,  A  and  H  the  tangent 

is  perpendicular  to  the  axis  of  abscissas,  and  one  of  these 
tangents  is  the  axis  of  ordinates. 
If  we  solve  the  equation 

T,X^  —  2x{/?  —  c)—l?C^O 
WQ  shall  have 


73- 


:r=- 


/^-^±V:^/?c+{d-c) 


but  y^l?c'-\-{l?—c)^</?-j-cj    hence   if  we  take  the   positive 


SINGULAR    POINTS. 


^57 


value  of  the  radical  part  the  result  will  be  less  than 
— — ,  that  IS,  less  than  "^b^  hence  it  will  give  no  point 

of  the  curve.  If  we  take  the  negative  value,  the  result  will 
be  numerically  less  than  —\cj  hence  there  will  be  two 
points  where  the  tangent  will  be  parallel  to  the  axis  of 
abscissas,  corresponding  to  the  point  on  that  axis  where 


b-c-W  ibc-{-{b-cY 
x^ 

3 

If  ^=o  the  equation  becomes 

ay"  =.v'* — bx^ 

in  which  case  the  oval   HA  is  contracted  into  a  conjugate 
point  at  A  as  in  Art.  154. 

If  b=^o  the  equation  becomes 

aj'''^  =^x^  -\-€X^ 
or 


'=±\/- 


•>      1  o 

A"^  -\-CX" 


a 


and 


dv  ;ix^-\-2CX 

^    ~      2\  ax'(x-\-c) 


T 


V 


In  this  case  the  curve  takes  the  form 

in  Fig.  74.     There  are  two  equal  values 

for  J'  with  opposite  signs  for  every  value       \f7^ 

of  X  on  the    positive   side  of  H  where         '^- 

dv 
x=^—c.     At  that  point  ^=  00,  and  the 

tangent  is  perpendicular  to  the  axis  of  ^'^'  ^^ 

abscissas.     If  we  make  "7~=o  we  have  jc=o  and  x=^  —  -z^  ; 

2C 

hence  the  tangent  at  A  and  at  T  and  T'  where  x^=^  —  —^  are 

3 
parallel  to  the  axis  of  abscissas. 


258 


DIFFERENTIAL    CALCULUS. 


If  we  make  both  b  and  c  equal  to  zero  we  have 

whence 

/  x^ 


and 


d\ 


In   this    case   the    curve   assumes  the 

form  in  Fig.  75.     There   is  no  negative 

value  for  x,  and  all  positive  values  of  ^ 

give  two  equal  values  for  j'  with  contrary 

dy 
signs.    At  the  origin  we  have  —7~  =0,  and 

hence  the  axis  of  abscissas  is  tangent  to 

both  branches  of  the  curve  which  is  a  cusp  of  the  first 

species 


Fig.  75- 


PART  IL 


Integral  Calculus. 


259 


I NTEGRAL     CaLCULUS. 


SECTION   I. 


PRINCIPLES  OF  INTEGRA  TION. 

(157)  The  problem  of  the  differential  calculus  is  to 
obtain  the  differential  or  rate  of  change  in  a  function  arising 
from  that  of  the  variable,  or  variables,  which -enter  into  it. 
The  corresponding  problem  of  the  integral  calculus  is  to 
pass  from  a  given  differential  of  a  function  to  the  function 
itself. 

The  first  of  these  operations  can  always  be  performed  i' 
directly  by  rules  founded  on  philosophical  principles.  The 
second  can  only  be  performed  by  empirical  rules  founded  on  ^ 
actual  experiment.  We  cannot  proceed  directly  from  the  dif- 
ferential to  the  function,  but,  as  it  were,  backwards ;  that  is, 
we  show  that  a  function  is  the  integral  of  a  given  differen- 
tial by  showing  that  the  latter  would  be  produced  by  differ- 
entiating the  former.  Thus  we  know  that  x'^  is  the  integral 
of  2xdx^  because  2xdx  has  been  shown  to  be  the  differential 
of  x'^.  Hence  the  rules  for  integration  are  merely  the  rules 
for  differentiation  inverted. 

While  rules  have  been  obtained  for  differentiating  every 
algebraic  function,  it  by  no  means  follows  that  every  differ- 

261 


262  INTEGRAL    CALCULUS. 

<ential  can  be  integrated.  The  number  of  simple  algebraic 
functions  is  very  small,  and  each  one  has  its  specific  form 
of  differential.  Should  any  function  be  complicated,  it  can 
be  analyzed  and  differentiated  in  detail,  applying  only  the 
rules  for  simple  forms.-  But  before  a  differential  can  be 
integrated,  it  must  be  reduced  to  one  of  the  forms  arising 
from  differentiating  a  simple  function ;  and  this  can  be  done 
in  comparatively  few  of  the  infinite  number  of  forms  that 
differentials  may  assume.  The  transformations  available  for 
this  purpose  form  one  of  the  chief  subjects  that  demand  the 
attention  of  the  student  of  the  integral  calculus.  The  dif 
ficulty  of  integration  is  very  much  increased  when  the  dif- 
ferential is  a  function  of  two  independent  variables,  for  the 
rate  of  change  in  such  a  function  can  give  but  little  indica- 
tion, generally,  what  the  function  is. 

There  is  still  another  difficulty  in  obtaining  the  exact 
integral  of  any  given  differential.  We  have  seen  that  the 
constant  terms  in  any  function  disappear  when  it  is  differ- 
entiated, and,  of  course,  when  we  come  to  integrate  an  iso- 
lated differential  expression,  we  cannot  know  what  constants, 
if  any,  should  belong  to  it.  In  such  a  case,  then,  we  pay  no 
attention  to  the  question  of  constants.  If,  however,  the 
function  should  occur  in  an  equation  we  can  generally  find 
from  the  conditions  expressed  by  it  what  value  would  belong 
to  the  constant.  Until  this  is  done  we  indicate  by  adding 
the  symbol  C  to  the  integral  that  a  constant  is  to  be  supplied 
if  needed  to  render  the  integral  definite.  Until  then  it  is 
said  to  be  indefinite. 

The  notation  indicating  the  integral  of  any  differential  is 
the  letter  s  elongated,  thusy^^/x would  be  read  "the  integral 
of  xdxT  This  notation  was  originally  adopted  by  Leibnitz 
to  indicate  the  stwi  of  the  infinitely  small  differentials  or  dif- 
ferences of  which  he  supposed  the  function  to  be  made  up, 
and  is  still  retained  as  a  matter  of  convenience  even  by 


FRINCiPLES    OF    INTEGRATION.  263 

those  who   reject  its  original   meaning,  as  employed  in  the 
system  of  Leibnitz. 

The  following  rules  for  integration  are  derived  from  those 
for  differentiating ;  being  in  fact  the  same  rules  inverted. 

(158)  If  the  differential  have  a  cofistant  coefficient  it  7?iay  be 
placed  loithoiit  the  sign  of  integi'aiion. 

For  we  have  seen  (Art.  lo)  that  the  differential  of  a  vari- 
able having  a  constant  coefficient  is  equal  to  the  constant 
multiplied  by  the  differential  of  the  variable ;  that  is  to 
say,  the  coefficient  of  the  variable  will  also  be  the  coeffi- 
cient of  its  differential;  hence  the  coefficient  of  the  differ- 
ential will  also  be  the  coefficient  of  its  integral,  that  is,  of 
the  variable ;  and  may  be  placed  outside  the  sign  of  integ- 
ration. 

Thus 

d{ax)^^ adx  hence  fadx  =  afdx  ==  ax 

(159)  The  integral  of  a  differential  fu7Ktio7i^  cottsisting  of 
any  number  of  terms  connected  together  by  the  signs  plus  and 
minus ^  is  equal  to  the  algebraic  sum  of  the  integrals  of  the  terins 
taken  separately. 

For  we  found  (Art.  9)  that  the  differential  of  a  polynom- 
ial is  found  by  differentiating  each  term  separately,  hence  to 
return  from  the  differential  to  the  polynomial,  which  is  the 
integral,  we  must  integrate  each  term  separately.     Thus 

d{x+y-z)  =dx+dy-dz 
hence 

J'{dx-]-dy—dz)=-Jdx-\-Jd) — J^dz=-x-\-y—z 

(160)  The  integral  of  a  monomial  differential  consisting  of  a 
variable^  multiplied  by  the  differential  of  the  variable  is  equal  to 
the  variable  raised  to  a  po7ver  with  an  exponent  increased  by  one., 
ajid  divided  by  the  increased  exponent  and  the  differential  of  the 
variable. 

Vv"e  have  in  (Art.  15)  the  rule  for  obtaining  the  differen- 
tial of  the  power  of  a  variable.     In  other  words  we   have 


264  INTEGRAL    CALCULUS. 

given  the  steps  by  which  we  pass  from  the  power  to  its  dif- 
ferential ;  and  hence  to  pass  back  from  the  differential  to  its 
integral,  that  is,  the  power,  we  must  retrace  each  step.  Thus 
in  the  first  case  we  diminish  the  exponent  by  one ;  in  the 
latter  we  increase  it  by  one.  In  the  former  we  multiply  by 
the  differential  of  the  variable ;  in  the  latter  we  divide  by 
it.  In  the  former  we  multiply  by  the  exponent  before  reduc- 
ing it ;  in  the  latter  we  divide  by  the  exponent  after  increas- 
ing it.     Thus 

Jnx^  ~~  '^dx = x"- 
because 

d{x'^)-=^nx"-^dx 
(131)   If  the  function  consist  of  the  power  of  a  polynom- 
ial multiplied  by  its  differential,  the  same  rule  will  apply. 
Thus  let  the  differential  be 

(ax-\-x"y^{a-\-2x)dx^=^{ax-\-x'^^'^d{ax-\-x'^) 


make 

then 

and 


ax-\-x^  '=-ic 
[ax -\-x^y^'  {a-\-2 x) dx = II ''" du 

EXAMPLES. 

x^dx  x'"^ 

Ex.  I.     What  is  the  intecrral  of  .'* 


JL 

Ex.  2.     What  is  the  integral  of  x^dx? 

dx 

Ex.  -i.     What  is  the  inte2;ral  of  — -p=} 

dx 
Ex.  4.     What  is  the  inte'j;ral  of  — -  ?  Aits.      — — i 

^  o  ^.o  2X' 


Ans. 

9 

4 

T,X^ 

Ans. 



4 

Ans. 

2\^'x 

I 

PRINCIPLES    OF    INTEGRATION.  265 

dx 
Ex.  K.     What  IS  the  integral  of  ax^dx-\-  — r=  ? 

ax"^  _ 

Ans.      — —  +  ^x 
3 

(162)   If  the  exponent  of  the  variable  in  the  case  pro- 
vided for  in  Art.  160  should  be  —  i,  the  rule  will  not  apply. 

For  by  this  rule 

.     .  x°       1 

/x~\/x^ — =— =  00 

^  00 

and  this  arises  from  the  fact  that  a  differential  with  such  an 
exponent  can  never  occur  under  the  rule  given  in  Art.  15  ; 
for  then  the  variable  must  have  been  jr°,  a  constant  quan- 
tity, that  cannot  be  differentiated.  Such  differentials,  how- 
ever, do  frequently  occur,  but  the  rule  for  their  integration 
must  be  drawn  from  a  different  source.     A\'e  have  found 

(ix 
(Art.  38)  that  the  differential  of  log.  x^ — =jc~Vjt:,   and 

hence  a  differential  of  this  kind  must  be  integrated  by  the 
rule  derived  from  that  given  for  differentiating  logarithms. 
That  is  to  say,  the  integral  of  any  fraction,  in  which  the 
numerator  is  the  differential  of  the  denominator,  is  the 
Naperian  logarithm  of  the  denominator. 

EXAMPLES. 

adx 

Ex.  I.     What  is  the  mtei^fral  of ?  Ajts.     ^  I02;.  .r 

^  X  ° 

2hxdx 

Ex.  2.     What  is  the  intesrral  of  — 7—, — s  .'' 

^  a-f-ox-' 

Ans.     log.  (^7-f  Av~) 

adx  a 

Ex.  3.     What  is  the  integral  of  —, — .''  Ans.     ~I  log.  x 

iix'-^'dx 
Ex.  4.     What  is  the  integral  of  — :^ — ?         -ins.     a  log.  x 

dx  . 

Ex.  ^.     What  is  the  intesjral  of  — , —  .''         Ans.     log.  (a-\-x) 


266  INTEGRAL    CALCULUS. 

(163)    ^f  the  differential  be  in  the  form  of  a  polymomial^ 

raised  to  a  poiver  denoted  by  a  positive  integral  exponent  atid 

multiplied  by  the  differential  of  the  variable^  the  integral  may  be 

fowid  by  expanding  the  poiver  and  nmltiplying  each  tertn  by  the 

differential  of  the  variable.      We  may  then  integrate  the  tenns 

separately.     Thus  let  us  take  the  expression 

{a-\-bx^"dx 

Expanding  the  binomial  and  multiplying  each  term  by  dx 

we  have 

a^dx-\-2  abxdx  -\-b^x^dx 

which  may  be  integrated  as  in  Art.  158. 

EXAMPLES. 

Ex.  I.     What  is  the  integral  of  (5  4-7^^)^^^^^^  ? 

Ans.     2^x-\-  '^^-x  ^  +  ^-^-x  ^ 
Ex.  2.     What  is  the  integral  of  {a-\-'^x'^^^dx'i 

Ans.  a^ X -i- 2,(1^ x^  -\---^-ax^  -\--i^-x'^ 
{ I  64)  Jf  (^  binomial  differential  be  of  such  a  form  that  the 
exponent  of  the  variable  withoitt  the  parenthesis  is  one  less  than 
that  of  the  variable  within.,  the  integral  ivill  be  fou7id  by  increas- 
ing the  exponent  of  the  binomial  by  one  and  dividing  it  by  the  new 
exponent  into  the  exponent  of  the  variable  within  into  its  coefficient. 
For  suppose  the  differential  to  be 

(a-^bx'^Y\x^^-\ix 
make 

then 

and 


a-\-bx^^  =^p 

dp^nbx''^~'^dx 
dp 

X^^'^dx^ — T 

nb 


from  which 


{a+bx""  Y^x'^-hlx  ——f 


PRINCIPLES    OF    INTEGRATION.  267 

of  which  the  integral  is 

{ni-{-\)7ib~     [m-\-i)nl? 
hence  the  rule. 

EXAMPLES. 

1. 
E.\  I.     What  is  the  integral  of  {a-\-bx")''^ ?nxdx} 

A  us.      — ria  +  bx'^\^ 

-1 
Ex.  2.     What  is  the  integral  of  {a'^-\-x^)    '-^xdx} 

Ans.     (,72+.r2)i 
3 
Ex.  3.     What  is  the  integral  of  {a-\-bx^)^cxdx} 

c{a  +bx^)^ 
Ans. } 

( !  65)  Every  rational  fraction,  which  is  the  differential  of 
a  function  of  x^  may  be  put  under  the  form 

A.T^^+B.v^^^-l  +  C.y"^-^+ D.r  +  E 

PV^  -\-Gx^-'^-{-¥Lx^-'^^ K^'+L'^^ 

in  which  the  greatest  exponent  of  the  variable  in  the  denom- 
inator exceeds  by  one  or  more  the  greatest  exponent  in  the 
numerator.  For  if  it  is  equal  or  less,  a  division  may  be 
made,  until  the  exponent  of  the  remainder  would  become 
less  than  that  of  the  divisor,  and  this  remainder  would  be- 
come the  numerator  of  the  fractional  part  of  the  quotient ; 
the  other  part,  consisting  of  entire  terms,  would  be  integrated 
as  in  Art.  159.  Hence  we  need  only  to  consider  the  method 
of  integrating  the  fractional  part  of  the  quotient,  or  rather 
any  fractions  of  the  form  already  given. 

(166)  For  this  purpose  we  resolve  the  denominator  into 
factors  of  the  first  degree  as 

{x — a){x — b){^x — c){x — //)  etc. 
and  place  the  fraction  under  the  form 

/    A  B  C  D 

( ^4- 7  4- + ,+  etc.)./.r 

^x—a     X — b     X — c     X — a  ' 


268  INTEGRAL    CALCULUS. 

in  which  A,  B,  C,  D,  etc.,  are  constants,  whose  values  are 
determined  by  reducing  all  the  fractions  to  a  common  de- 
nominator, and  placing  the  sum  of  the  numerators  equal  to 
the  original  numerator  (the  denominators  being  identical). 
Since  this  equality  of  the  numerators  must  exist,  indepen- 
dent of  any  particular  value  of  x,  the  coefficients  of  the  like 
powers  of  x  must  be  respectively  equal  to  each  other;  and 
this  will  furnish  enough  equations  to  determine  the  values 
of  the  constants.  Substituting  these  values  the  fractions 
may  then  be  integrated  separately. 

(167)  For  example  let  us  take  the  fraction 

which  by  decomposing  the  denominator  may  be  put  into  the 
form  2adx 

{x-\-a){x—d) 
which  we  transform  into 

A  B 

^x-\-a     X — a' 

which  being  reduced  to  a  common  denominator  becomes 

Kx—Ka-\'Bx-{-Ba 
{x — a)[x-\-d) 
Making  this  last  numerator  equal  to  that  of  (i)  we  have 

2^=A.r — Aa-\-V>x-\-^a 
or 

(A+B):r  +  (B-A-2)^  =  o 
from  which  we  obtain 

A+B=o 
and 

B— A  — 2=0 
whence 

A  =  — I   and  B  =  i 
Substituting  these  values  of  A  and  B  in  (2)  we  have 

2a;^x  ilx  dx 

x^ — a-     X  —  a     x-j-a 


[—7-  + K^  (2) 


PRINCIPLES    OF    INTEGRATION.  269 

and  by  integration 

/2adx         r  dx         r  dx 

( I  68)   Let  us  next  take  the  fraction 

a^  -\-bx" 


a'^x  —  x^ 


ix 


in  which  the  factors  of  the  denominators  are  x  and  {cr  —x~) 
or  x{a-\-x^{a — x^.     If  we  make 

a^-\-bx'^  ABC 


x{(i-\-x){a  —  x^      X     a — x     a-\-x 
and  reduce  the  second  member  of  the  aquation  to  a  com- 
mon denominator  we  have 

a^  +bx^-      Aa"  -Ax-  +Bax+Bx-  -j-Cax—Cx" 
a'^x  —  x'^  x{a  —  x^{a-\-x^ 

and  placing  the  coefficients  of  the  like  powers  of  x  in  the 
numerators  equal  we  have 

B-A-C=/; 

B^+C^=o 

A^-=^3 

The  last  of  these  equations  gives 

A— ^ 
which  reduces  the  first  to 

B-C=^+^ 
and  this  combined  with  the  second  gives 

a-\-b  a-\-b 

B  = and  C  =  — 

2  2 

Substituting  these  values  of  A,  B  and  C  in  equation  (i)  we 
have 

a^ -\-bx'^  adx         a-\-b  a-{-b 

— ^ "o  dx = + ~7  \dx  —  ~7     ,     \dx 

a^x—x^  X        2\a  —  x)  2\a-\-x) 

and  by  integration 

^a'^-\-bx-  a-^-b  ^         .  a-\-b  r     ,     \ 

J  ^2^_  ^.,dx=a\og.  x--^  log.  {a-x)--^  log.  {a+x) 


270  INTEGRAL    CALCULUS. 

which  may  be  reduced  to 

a  log.  ^—(^+^)  log.  {a^—x'^^ 

Note. —  The  second  term  of  the  integral  must  be  negative  ;  for  since  d{a—x)  is. 

d.x                             dx 
— </ji-,  weshall  have  <^(log.  («— jir)=— — — — 7  and  hence  -must  be  the  differential 

of  —log.  {a—x). 

(169)  Let  us  now  take  the  fraction 

xdx 
x'^  -\-4ax—d^ 
To  find  the  factors  of  the  denominator  we  must  make  it 
equal  to  zero,  and  solve  the  equation  which  gives 
x  =  —  2a±\'^^2^^2 

and  hence  the  factors  of  the  denominator  will  be 


and 

x^2a—^^a2^^2 

To  simplify  the  expression  we  will  represent  the  constant 
part  of  each  factor  by  E  and  F  and  we  shall  have 

x^  +4ax-^^  ~{x  +  E)(x-\-F) 
and  we  may  make 

X  A  B         Ajc+AF+B.t+BE 


oc'^-j-  4ax—3^     x-^-Fi     x-\-F  {x-{-E)(x-i-F) 

making  the  numerators  equal  we  have 

A^+ AF +BJC+BE  =jc 
whence 

and 

from  which 


A+B  =  i 
AF+BE=o 


E  T^ 

A  =  T^ — t:;  and    B 


E-F  "-"^    ^~     E-F 
Substituting  these  values  of  A  and  B  we  have 

Xi/x  E      r    dx  F      r    dx 


/xdx         __     hi      r    dx  ^       C 

x^  +  Aax—l?'^~F—FJ  x-\-  E~E—fJ 


PRINCIPLES    OF    INTEGRATION.  27  1 

which  becomes  by  integrating 

E  F 

jT^plog.  (x  +  E)-g^3|;log.  (.r  +  F) 

or  by  substituting  the  values  of  E  and  F 


xdx 

2 

cr-hV,^ 

r  +  ^2 

X- 

'  -\-4ax— 

2(1 

2V,4il- 

+  P 

1-'  -{-b' 

1^,,    /., 

log.  {x-{-2a-\-V ^a''^ -\-b'^) 


^/      o    ,    ;2      '^og.{x-\-2a-y  4a-+b-)  J 

(170)  In  all  these  cases  the  factors  of  the  denominator 
are  unequal.  If  a  part  or  all  of  them  are  equal  the  rule 
will  not  apply.     For  suppose  we  have 

{x — d){x — b){x — c){x — d){x — c) 
which  we  make 

A  B  C  D  E 

x — ax — b     X — ex — d    X — e 
if  some  of  these  factors  are  equal,  say  a^=b^=^c,  we  should 
have 

P^^+etc.  A+B+C       D  E 

: :^ ! ! I   I  

(.r — a)^{^x — d){x~e)  x — a  x—dx — e 

Thus  in  reducing  the  second  member  to  a  common  de- 
nominator, A  +  B  +  C  would  have  to  be  considered  as  a  sin- 
gle constant  A',  and  the  three  constants  A',  D  and  E  would 
not  be  sufficient  to  establish  the  five  equations  of  condition 
which  are  required  in  making  equal  the  coefficients  of  the 
like  powers  of  x.  In  order  to  avoid  this  difficulty  we  decom- 
pose the  original  fraction  and  make 

P:y^+Q^v^+  etc.    _A+Bx-f-ar^        D  E 

{x—a)'^{x—d){x—e)  (.T — a)'^  x—d    x—c 

which  contains  the  necessary  number  of  constants,  and  at 
the  same  time,  when  reduced  to  a  common  denominator, 
will  produce  a  numerator  containing  x  to  the  fourth  power; 
thus  giving  a  sufficient  number  of  equaaons  between  the 
coefficients  of  the  like  powers  of  x. 


272  INTEGRAL     CALCULUS. 

In  the  meantime  the  expression 

A+B.v  +  Cjt:^ 
{x—a)^ 
may  be  put  into  the  form 

A'  B'  C 


{x—a)^  {x—a)^  x—a 
are  determinate  consta 
X — a^^u  then  x^=^u-\-a 


in  which  A',  B  ,  C'  are  determinate  constants.     For  let 


and 

A+B:r+C^2     A-f  B^  +  C^2  _^b?/  +  2C^?/4-C?/' 


{x  —  ci)'^ 


A  +  B.?  +  C^3     B  +  2C^     C 

+■ 


-2 


and  replacing  the  value  of  u  we  have 

AH-B.r+Cj^s     A+B^+C«3     B  +  2C^       C 


[x — d)"^  [x — d)^      ~^  {^x — a)"^     X — a 

and  since  these  numerators  are  constant  we  may  represent 
them  by  A',  B',  C',  which  gives 

K-^-^x^Qx^  A'  B'  C 

_ — „  :— -I  I  

{x  —  a)^  (x—a)'^     {x—a)^     x  —  a 

which  is  the  required  form. 

As  this  demonstration  may  be  applied  to  an  expression 
containing  any  power  of  x,  we  make  the  proposition  a  getl- 
eral  one,  that 

p^m-l_Q^m-2^ R^  +  S 


< 


(X-C7)»' 

A  A'  A"  ,      ^ 

4-  etc. 


{x-a)^~^{x-a)^^^  '^{x-aY'-^ 

Hence  to  integrate  the  expression 

Vx^-hQx^+etc. 

{x—a)^{x — d){x—e) 

we  write 

/      A  A^  A"         D  E    \ 

\{x—a)'-^{x—a)^     x—a     x—d    x—eJ 


PRINCIPLES    OF    INTEGRATION. 


273 


and  reduce  these  fractions  to  a  common  denominator  and 
find  the  values  of  A,  A',  A'',  D,  E,  in  the  manner  already 
stated.  We  shall  then  have  to  find  the  integrals  of  the  fol- 
lowing expressions, 

E  _D_         _A^  A'  A 

X — e         X — a         X — a     '    \x — ay     '    (.r — ay 
the  three   first  we  can  integrate   by  the  rule  for  logarithms 
and  the  others  as  follows. 

Since  dx  is  the  differential  of  .r — ^  we  will  make  x  —  a=-z  ; 
then  we  have 

C    XiLx  /» Kdz       r        .    y  A  A 

J  \x—aY      J     z^       ^  2z~  lyx — a)" 

and 

f  A'dx    _  fA'dz_      A^_       A' 
J  {x — d)^     J    z'-^  z  x—a 

Hence 

/»     P.y^+Q^y^+etc.         _  A  A^        ^ 

^  {x—d)'^{x—d){x—c)  2{^x—d)'^     x—a 

+A"log.  (jt:-^)  +  Dlog.  (.T-./)  +  Elog.  {x-e) 

(171)   Let  us  take  for  example 

x~dx 
x'^  — ax^'^ — a"x-\-d-^ 
the  denominator  of  this  fraction  may  be  resolved  into  the 
factors 

{x"  —a'^'){x—d)^^{x—d){x-{-d)(x — a) 
or 

{x — ay{x-\-a^ 
Making  then 

x^  A  M  B 

{x-aY{x-\-a)~{x-a)'-^  x-a^  x^-a  (i) 

and  reducing  the  second  member  to  a  common  denominator, 
we  have 

x^  A(.T+^)+A^(a-^-.?")  +  B(.T-^)^ 

(jr — d)^{x-\-d)~  {^x — a)- {x  ^  ij) 


//' — =  - 


^—  — ±-lI^     -^ — ^^ ^J2£       - — C 


!17!E    -Z  alzii^  farms  IE  Z3if:» 


—       +  -,- 


'w± 


js. — i:>r 
:x — r=^z.  ~TerL  jr=r — r  ^mf  zsr^rx. 


:  /••—  /— 


=^11^1 — — — :—     ^     z — s^"- 


J  173     '^CSse"  '  -. -z^-^  :_^:-:_i^:  2EC  CTiaL  ' .   ^^    i 

•wiH  ■y~Tr>^"  Tier  'i-i-i-.  ... 

2Jrj  tniTJ   Z-.  '  z  -    z  IZ  ~Jtt  ^KgHTH-  ZxCc:..  ITLC  TTgJ  ^Fol   Ze 


Trrrm  HE  tlEierS'  le  i^ 

Ti   fc  ~^'~  "Vr  lifxZ.  ~re-  zri5s::s:::nni;2C=' 
— .f^— "'e~  ;f  tJi-r  — T'^g'-^  eoimiiiiL.  "snoii  su 
ciSsrsikis-  3E  2iET_     T^n5  :s  Tn..  t:^~  ire  ■  -  ;    -  ■  ---_-_ 

1  Ttri^T.^  sziisiS'  lie  d  '    ' '      '  '  "^ 

ti'T  -i'tsc  FTi~  ~Tr°~  -jx  "ire  "  k . 


^F%.  ri         T^;  :  lie  s^EiBiie  ex  inK  — ;r.T'.5^e  is 

Jrrr^^  KAtTL   (leVc    X  ~eiZL£   "Tie  ~Tr)i^\-  ,lf  TJe  ■ETlg'-e  •> i_^.    I^lI. 


276 


INTEGRAL    CALCULUS. 


ax' 


+c 


(l) 


Now  to  determine  the  value  of  C  we  give  to  Jt:  a  value  cor- 
responding to  a  known  value  of  S.  But  we  know  that  at 
the  origin  in  A  where  ^=0,  we  have  also  S=o,  and  by  sub- 
stituting these  values  in  equation  (i)  we  have 

o=o+C  hence  C=o 
and 

S= — 


is  the  definite  integral. 

If  now  we  wish  to  know  the  value  of  any  specific  part  of 
the  triangle,  such  as  ADD',  we  make  jr=jf'=AD,  and  we 
have 

ax"^      xy      ADXDD' 
22  2 

This  is  the  specific  integral. 

( 1 74)  We  are  not  bound,  however,  to  make  the  value  of 
S  commence  at  the  origin  where  a^=o.  We  may  if  we  choose 
estimate  it  from  any  line  as  DD'. 
jt:=AD=jc')  we  should  have 


In  this  case  (making 


ax 


o=" 


■fC 


whence 


ax 


AD 


—  a- 


2  2 

and  substituting  this  value  in  equation  (i)  we  have 

ax^         AD^ 

S= — a 

2  2 

This  IS  again  the  definite  integral.  For  any  portion  of  the 
triangle  estimated  from  DD'  we  give  the  corresponding  value 
of  x^  say  x^^h.Yj'=^x" ^  which  gives 


PRINCIPLES    OF    INTEGRATION.  277 

o_^/  vo       'OX      ae'-ad 

2  2 

for  the  value  of  the  area  DD'E'E. 

( I  75)  There  is  another  method  of  disposing  of  the  inde- 
terminate constant,  which  consists  in  giving  to  the  variable 
two  definite  values,  and  then  subtracting  one  integral  from 
the  other.  This  is  called  integrating  between  limits.  Thus 
ir  the  case  last  noted,  if  we  make  x  successively  equal  to 
x'  -  -AD  and  x"=^AE,  we  shall  have 

ax  "  „     ax  " 

S'= +C  andS"= +C 

2  2 

and  subtracting  the  first  equation  from  the  second  we  have 

2^  ^  2 

the  constant  C  having  disappeared  in  the  subtraction. 

The  notation  for  this  kind  of  integration  consists  in  plac- 
ing the  two  values  of  the  variable  at  the  extremities  of  the 
sign  of  integration  ;  thus 


/ 


X" 

axdoo 

X' 


indicates  that  the  integral  is  to  be  taken  between  the  two 
values  of  x  represented  by  x"  and  x  ;  the  subtractive  one 
being  at  the  lower  extremity  of  the  sign ;  and  the  integral 
would  be 


ox  ^     ax 


When  the  integral  is  to  be  taken  for  any  particular  value  of 
.r,  as  x\  it  would  be  written 


,/  x=x' 


axdx 


which  indicates  that  the  integral  is  to  be  taken  where  x=x\ 


278  INTEGRAL  CALCULUS. 


EXAMPLES. 


Ex.  I.     Integrate  2:^^.;^ between  the  values  oix^^a  and  x^=^b. 

Ans.     b--a^ 

Ex.  2.     What  is  the  integral  of    /    3^^^/jt:.         A7is.     b^—a^ 

—x"i/x.  Ans.     t(^^ — a^) 

Pb 

Ex.  4.     Integrate   /    2{e-{-x)dx.         Ans.     b"-{-2e{b — a)— a" 

fb 
Ex.  5.     Integrate   /     2){.^-\-Jix^)^2nxdx. 

Ans.     {e-{-nb^)^  —  {c-\-na")^ 

/^  dx  e-\-  b 

— ^ — .  Ans.     log.  — r 


a 


(176)  INTEGRATION    BY    SERIES. 

If  it  be  required  to  integrate  a  differential  of  the  form 

'^{pc)dx  in  which   F(:^^)  can  be  developed  into  a  series,  the 

approximate  integral  may  often  be  found  by  (Art.  164),  and 

if  the  series  is   rapidly  converging,   its   true  value  may  be 

nearly  reached.     Let  the  differential  be 

dx 
- — i — ^=^\\-\-x^)'~^dx 
\-\-x^      ^  ' 

developing  by  the  binomial  theorem  we  have 

and  multiplying  by  dx  and  integrating  we  have 

/dx  x"^      x^      x"^ 

— , — 2—^— — + — — — +  etc. 
i-j-x^  3        5         7 

EXAMPLES. 

Ex.  I.     What  is  the  integral  of  — 7^ — ?  Ans. 


PRINCIPLES    OF    INTEGRATION.  279 


dx 

Ex.  2.     What  is  the  intecfral  of  ?  Atis. 

°  a  —  X 

dx 
Ex.  x.     What  is  the  integral  of  7 r^  ?  A71S. 

dx 
Ex.  4.     What  IS  the  integral  of  — -^=^  ?  A71S. 


(177)      INTEGRATION  OF  DIFFERENTIALS  OF  CIRCULAR  ARCS. 

We  have  seen  (Art.  47)  that  if  ?/  designate  the  sine  of  an 
arc,  then  the  differential  of  an  arc  will  be 

du 


hence  the  integral  of  the  function  of  the  form 

dx 


V  i-x^ 
will  be  an  arc  of  which  x  is  the  sine. 
(178)   If  the  expression  is  of  the  form 

dx 


V  a^  -  x^ 
we  may  make  x==a7>  then 

x'-^—a'^v^  and  a^—x''^=a'^—a'^v^=a^{i  —  v'^) 
and 

dx=^adv 
whence 


dx  adv  dv 


and 


V d-  —  x^     <-W  i—v^     ^i-v 

n        dx        _  r      dv 

^   V  a-—x-     ^  V  I— z/2 

which  is  an  arc  of  which  t'=~  is  the  sine. 

a 


28o  INTEGRAL    CALCULUS. 

(179)  If  ^  represent  the  cosine  of  an  arc,  then  the  dif- 
ferential of  the  arc  (Art.  47)  will  be 


hence  the  integral  of  the  form 


(IX 


V  i—x^ 
will  be  an  arc  of  which  x  is  the  cosine. 
If  the  expression  is  of  the  form 


cLx 


^  a--x^ 
it  may  be  integrated  as  in  (Art.  178) 

(180)  If  7  represent  the  tangent  of  an  arc  then  (Art.  47) 
the  differential  of  the  arc  will  be 

dv 


hence  the  integral  of  a  function  of  the  form 


ion 
dx 


I  -\-x^ 
will  be  an  arc  of  which  x  is  the  tangent. 

(181)  If  the  expression  is  of  the  form 

dx 
a^  -\-x^ 
we  may  make  x^=^av^  whence 

dx^=^adv  and  a^  -\-x'^  '=-a^  -^-a^v'^  ^=^a^{^i-\-v'^^ 
whence 

r     dx     _  p       adv       _  i  /*    d-c^ 

^  a^ -\- x^     ^  a^  {\-\-v'^^      d'^  i-\-v^ 
which  IS  equal  to  —  into  an  arc  of  which  v——  is  the  tang-ent. 

(182)  If  we  represent  the  versed  sine  of  an  arc  by  z  we 
have  (Art.  47)  for  the  differential  of  the  arc 

dz 

V  2S Z^ 


PRINCIPLES    OF    INTEGRATION.  251 


hence  the  integral  of  a  function  of  the  form 

(Ix 


V  2X  —  x^ 
will  be  an  arc  of  which  x  is  the  versed  sine. 
(183)  If  the  expression  be  of  the  form 

dx 


^  2ax  — x^ 
we  may  assume  x^=-av^  whence 

dx^=-adv  and  2ax—x'^  becomes  la^v—a^v^ 
or 

whence 

/dx  I*       adv  /•      dv 

which  is  an  arc  of  which  z^=—  is  the  versed  sine. 

a 


SECTION  II. 


INTEGRA  TION  OF  BINOMIAL  DIFFERENTIALS, 

( 1 84)  The  general  expression  for  a  binomial  differential 
may  be  reduced  to  the  form 

x^-^a^-bxf'Ydx  (i) 

in  which  m  and  n  are  whole  numbers,  7i  is  positive  and  x 
enters  but  one  term  of  the  binomial. 

For  if  m  and  n  are  fractional  we  may  substitute  another 
variable  with  an  exponent  equal  to  the  least  common  mul- 
tiple of  the  denominators  of  the  given  exponents,  which  will 
then  be  reducible  to  whole  numbers. 

If,  for  example,  we  have 

x^^{a-\-bx^^)^^dx 
we  make  jr=2^,  then  dx^=-bz^dz^  and  we  have  by  substitu- 
tion 

Z 

(iZ^(a-\-bz^Ydz 

If  n  is  negative  we  can  make  x^—^  and  the  expression 

would  become 

x^-^{a^-bx-'')'^dx  =  -z-'''^^{a^bz''y^dz 
in  which  the  exponent  of  z  within  the  parenthesis  is  pos- 
itive. 

282 


INTEGRATION    OF    BINOMIAL    DIFFERENTIALS.  283 

If  the  expression  is  of  the  form 

we  may  divide  the  terms  within  the  parenthesis  by  x''' ^  and 
multiply  liie  parenthesis  l)y  it,  thus 

or 

w  P_ 

^  ^"^  q{a-\-bx^-''Ydx 

thus  we  may  secure  the  three  stated  conditions. 

P  .... 

(185)  If  —  is  a  whole  number  and  positive,  the  binom- 

ial  may  be  expanded  into  a  finite  number  of  terms  and 
integrated  by  Art.  (163).  If  it  is  entire  and  negative  the 
function  becomes  a  rational  fraction. 

EXAMPLES. 

Ex.  \.     Integrate  the  expression 

x'^{^a-\-bx^^'^dx 
Expanding  the  binomial  we  have 

a^x^  dx  +  2  abx  ^  dx  -\-b"x^dx 
and  integrating  each  term  separately  we  obtain  for  the  in- 
tegral of  the  binomial  differential 

a~x'^     abx'^     b^x^ 
fx^{a-\-bx^Ydx-=^—:r-V—-^—-- 

Ex.  2.     Integrate  the  expression 

x'^{a  •\-bx'^^^dx 

Ex.  3.     Integrate  the  expression 
x^{^a-\-bx^^^dx 

a^x^       ^aHx^      %abKx^^      bKx-^^ 
5      '        b         '        II  14 


284  INTEGRAL    CALCULUS. 

Ex.  4.     Integrate  the  expression 

x'{a^-b~x^Ydx 

a^x^      yi-b-x^''      T^ab^x^'^     b^x"^^ 
Ans.      —7—+ + —  +     7^~ 

6  10  14  TO 

(186)  Every  binomial  differential  may  be  iiite grated  when  the 
exponent  of  the  variable  without  the  parenthesis.,  increased  by  one., 
is  exactly  divisible  by  the  exponent  of  the  variable  within. 

To  effect  this  we  substitute  for  the  binomial  within  the 
parenthesis,  a  new  variable  having  an  exponent  equal  to  the 
denominator  of  the  exponent  of  the  parenthesi:-,  thus  in 
the  expression 

x'^-'^(a^bx^)^dx  (1) 

we  make 

a+bx''=z'i  (2) 

then 

{a-\-bx^Y=zP  (3) 

From  equation  (2)  we  have 

'z'i—a\''' 


X- 


/z'J  —ay" 
V     b     ) 


and  raising  both  members  of  the  equation  to  the  ?nth  power 
we  have 

(z'i  —a\  "■ 
Differentiating  and  dividing  by  m  we  have 

m 

.,»-V,=^-(5^f''  s^-V.  (4) 

and  multiplying  together  equations  (3)  and  (4)  we  have 


x»'-^{a  +  bx^)'^dx=^zP+^-\—^) 


dz 


ni  .  .  . 

If  now  —  is  an  entire  positive  number,  this  expression  may 


INTEGRATION    OF    BINOMIAL    DIFFERENTIALS.  285 

be  integrated  by  raising  — i —  to  a  power  consisting  of  a 

limited  number  of  terms,  and  each  term  can  be  integrated 
separately. 

171 

If  -ji  be  negative  we  may  l)y  formula  D  (Art.  211  )  increase 
the  exponent  until  it  become  positive. 

EXAMPLES. 

( I  87)   Integrate  the  expression 

x'^  {a -\- bx^'-y'^  dx 

a-\-bx'  =3" 

(^  +  ^;c-)-=2;^  (i) 

—JT-  (^) 


Assume 
then 


X' 


an' 


b 
Multiplying  (i),  (2),  (3),  together  we  have 


zdz  .  , 

xdx——r  (3) 


—  a 


x''{a^bx^Ydx  ^z^—^dz 


of  which  the  integral  is 

z^      az^      {a^bx-Y     a(a-\-bx'^)^ 


(188)  Integrate  the  expression 

1 
x^ia+bx-y^^dx 

Make 

a-j-bx'  =z^ 

then 

{i7-^bx~y-=z  (i) 


286  INTEGRAL   CALCULUS. 

„      Z^—  a 

and 

zdz 
xdx=-y  (3) 

Squaring  (2)  and  multiplying  by  (i)  and  (3)  we  have 

„,             -1           ,z^  —  a\^  z^dz      z'^dz—2az^dz-\-a^z'^dz 
x^ia-^bx^^Ydx^^—j—)   -Y~= J, 

of  which  the  integral  is 

z^       2az^     a^z^ 


7^^       5^3  -r  ^^3 
and  restoring  the  value  of  z  we  have 

r  u   _j_.   2^1v       {^+1^^^')^     2a{a-\-bx^Y     aH^a±bx^ 
fx^{a+bx^Ydx= ^^^ -^^ + ^^ 

(189)  Integrate  the  expression 

x^{a-\-bx^Ydx 
Make 

a-{-bx'^^^z^ 
then 


2 


(a+bx'-y^z^  (i) 

also 

z^ — a 

and 

3z^dz 
2xdx= — 7 —  ( 3 ) 

Multiplying  the  square  of  (2)  by  (i)  and  (3)  we  have 
x^{a+bx^Ydx=^[^—^)  2V2 
3^* 

3s"'  ^//s       ■Ty^'^adz      T^a^z^dz 

~     2b'^     "     b'^      "^     2^3 


•    INTEGRATION    OF    BINOMIAL    DIFFERENTIALS.  287 

which  being  integrated  is 


22^3       8^3  -r  jQ^3 

Substituting  the  value  of  z  we  have 

/     /          ,x4     _3(^+/^^2)V-     3^(^+/^^2p     3«2  (^4.^^2)1 
Jx^ia.dx^)  dx- ^^^^ ^^—4- ^3 

( I  90)  Integrate  the  expression 

_i 

x^{a-\-x'^^   ^dx 

Make 

a-\-x^^=^z^ 
then 

x^^^z^—a  (i) 

xdx^=zdz  (3) 

Multiplying  together  (i),  (2),  (3),  we  have 

_j_ 
x'^dx{a-\-x^)    ^=^(z^—a)z~'^zdz=  {z^—a)dz 

and  integrating  we  have 

z^  (a-j-x^)^^  ^1 

— --az  — —  a{a  -\-  x^)^ 

3  3 

(191)  Integrate  the  expression 

x^{a^  -{-x")~'^dx 
Make 

a"  -f-x"  — z 
then 

x^=^z—a^  (i) 

2xdx^dz  (2) 

and 

Multiplying  together  (2),  (3)  and  the  square  of  (i)  we  have 

(5_^2\2^-l  /^ 

x^ia'^  -f-x" )    ^dx= 

'  2 

which  being  expanded  becomes 


288  INTEGRAL    CALCULUS. 


--^dz=- — -J- 


2  2  2  2Z 

Integrating  and  substituting  the  value  of  z  we  have 

4  2 

(192)  Another  condition  under  which  a  binomial  differ- 
ential may  be  integrated  is  as  follows : 

Put  the  expression  x^~\a-\-bx'^Y dx  into  the  following 
form 

oo"'-\{^-~{,-\-b)x'''Y<i^ 
or 

■  x^-'^x^  f^-\-b)'Jdx=x        ^       {ax-^'-^-bYdx 
By  (Art.  i86)  this  expression  is  integrable  when 

q       in      p 


n  n       q 

is  a  whole  number,  hence 

A  binomial  may  be  integrated  when  the  expoiient  of  the  vari- 
able without  the  parenthesis^  increased  by  ojie,  divided  by  the 
exponent  of  the  variable  within  the  parenthesis^  and  added  to  the 
exponent  of  the  parenthesis  is  a  whole  number. 

EXAMPLES. 

( 1 93)  Integrate  the  expression 

a{\-\- x"^^    '^dx 
Make 

v'^x^  =  1  -\-x'^ 
then 

(i+.T-^)"^=7'-3^-3  (i) 


INTEGRATION    OF    BINOMIAL     DIFFERENTIALS.  289 

also 


2__i 


v'^  —  i 
whence 

_   —vdv 

and 

i=x^{v^--iY  (3) 

Multiplying  together  (i),  (2),  (3),  we  have 

3.  adv 

a\\-{-x'')    "dx^-  —  -;^ 

and  by  integration 

adv      a  ax 


6^-  7'        V     I  -^-X"' 

( I  94)  Integrate  the  expression 

L  ^/jc 

dx\a"-\-x'^)    ^=- 


Make 


z;— a^  +  V  t^-^-jc- 
then 

xdx  x-\-V  a'-^-hx'^ 

dv=dx+  .,=: —  ^JC 

hence 

dv  dx 


Representing  the  integral  sought  by  X^  we  have 

^   r        dx  r  dv  ,—r, 5, 

( I  95)   Integrate  the  expression 

x"dx 

's/a-  +x- 

Representing  the  integral  by  Xo  we  have 

x^dx 
dX,=—-==^ 
V  a~  H-Jt" 


290  INTEGRAL    CALCULUS. 

Make 

then 

a^xdx-{-2x^t/x  a^dx  2x'^dx 

dv^=-- 


hence 

dv=a^dXQ-j-2dX2 

where  Xq  has  the  same  value  as  in  (Art.  194).     From  this 

we  have 

d?^      a"dX.n 

dX.= — 

^2  2 

or 

_Z^  «^Xn 

Xo  ■ 


2  2 

Replacing  the  value  of  Xq  and  v  we  have 

(196)  Integrate  the  expression 
Make 

Z;2_^2— j_^2 

then 

x~^=v^-i-i  and  ji£:~*=(z^2  +  i)2  (i) 

also 

x^={v^-\-j)   ^  whence  dx=^  —  {v^-i-i)    ^vdv  (2) 

and 

Multiplying  together  (i),  (2),  (3),  we  have 

a:~*(i— jv^)   ^^jc=  —  (z^^ +i)^(z;^ +1)   ^ z^^z/=-(z/2  +  i)^2» 


INTEGRATION    BY    PARTS.  29I 

Integrating  we  have 

I  -\-2X^ 


fx~^{i—x^)   ^dx=-  —  —  —  27=— \ — Vi— .r^ 

(197)  If  a  binomial  cannot  be  integrated  by  any  of  these 
methods,  there  are  others  to  which  we  may  resort.  These 
consist  in  making  such  a  transformation  of  the  expression 
that  the  exponent  of  the  variable  without  the  parenthesis 
or  that  of  the  parenthesis  itself  may  be  reduced  so  as  to 
bring  the  differential  into  one  of  the  integrable  forms.  This 
is  done  by  separating  the  differential  into  parts,  one  of 
which  shall  be  an  integral  quantity,  and  the  other  the  form 
to  which  we  desire  to  reduce  the  expression.     This  is  called 

INTEGRATION    BY    PARTS. 

To  effect  this  we  resort  to  the  principle  on  which  the  pro- 
duct of  two  variables  is  differentiated.  We  have  seen  (Art. 
11)  that 

d{trt^  =  udv + vdu 
hence 

uv  =-J'udv  -\-fi  'die 
or 

/"udv = uv  —fvdu  ( I ) 

If  now  we  can  so  transform  the  binomial  differential,  that 
while  it  is  represented  by  the  first  member  of  the  equation 
(i),  it  may  also  be  represented  in  its  transformed  state  by 
the  second  member,  we  see  that  the  integral  may  be  made 
to  depend  on  that  part  represented  by  vdu^  and  ihai  may  be 
made  to  assume  in  certain  cases  the  form  of  an  integrable. 
differential. 

The  two  general  methods  of  doing  this  are,  either  to  make 
the  part  represented  by  du  to  contain  the  variable  without  the 
parenthesis  with  an  exponent  diminished  by  that  of  the 
variable  within  the  parenthesis ;  or  else  to  contain  the  par- 


292 


INTEGRAL    CALCULUS. 


enthesis  itself  with  an  exponent  diminished  by  one.  In  all 
other  respects  this  part  is  to  be  identical  with  the  given  dif- 
ferential binomial. 

The  following  is  the  first  of  these  methods. 

For  convenience  we  represent  the  exponent  of  the  par- 
enthesis by/,  which  is  supposed  to  represent  a  fraction  ;  and 
substitute  m  for  in—\  ;  and  we  have  the  general  form 

in  which  m  and  n  are  whole  numbers. 
Make 

in  which  s  may  have  any  required  value.  Differentiating 
we  have 

dv=-biisx'^~\a-\-bx  ^^  y~h/x 

If  we  now  assume 

x^  {a-i-bx'^)P  dx  ^  udv 
we  have 

x''^  {a-\-  bx  ^  )  -?^  dx 


u^= 


or 


?/=:■ 


bnsx''^~\a  -\-  bx^^Y~^'^^^ 
bns 


which  being  differentiated  gives 

C  (in—n-\-\)x^-''{a-\-bx''')'P-^^^ 

\       du'= } dx 

J  b/is 

^  {p-s-^i)x''''(a+bx''')P~' 


-dx 


but 


hence 


(a+bx^')P-'-^'^  =  {a^bx'')P-'{a-\-bx''') 
=  a{a^bx")P-^'-\-l?x'^(a+lKX'')P-' 


du 


=[ 


a{m  —  n-\-\)x'"^'''^     (w  + 1  +/// — ns)x'' 


bns 


ns 


{a^bx'^)i'-^dx 


INTEGRATTOX    RV    PARTS.  293 


If  now  we  take  the  value  of  s  such  that 
we  have 


and 


m-\-i 


du = /,     .  , , — X dx 


Substituting  these  values  of  z/,  z',  du  and  dv  in  equation  (i) 
we  have 

Formula  A 

i  fx'"\a-{-hx''Ydx  — 

-*      x^-'^'^\a-\-bx''Y^^~a{in  —  n-^\)fx^-^{a-^bx''Ydx 
«  /'(///  +  /;/  +  !) 

in  which  we  find  the  integral  of  the  given  differential  to 
depend  on  the  integral  of  a  similar  differential  in  which  the 
exponent  of  the  variable  without  the  parenthesis  is  dimin- 
ished by  that  of  the  variable  within  it. 

In  like  manner  we  should  find 

fx^-''{a-\-hx''Ydx 
to  depend  on 

fx"^-'^^{a^-hx'^)Vdx 
and  we  may  thus  continue  to  diminish  the  exponent  of  the 
variable  without  the  parenthesis  as  long  as  it  is  greater  than 
that  of  the  variable  within  it. 

(198)  There  is  frequent  occasion  to  integrate  binomials 
of  the  form 

x'^dx 


V  a'~—x- 
Representing  its  integral  by  Xm  we  have 

_  r    x"^./v 
''  Va-  —  x- 


294  INTEGRAL    CALCULUS. 

and  substituting  in  the  formula  A  (Art.  197) 

—  I  for  <5 

2  for  n 

a^  for  a 

-\ioxp 

we  have 

FORMULA   a. 


X„,=  /  ;:= /  ,  — y  a^ — X^ 


(199)  Integrate  the  expression 

adx 


's/  a^  —  x^ 

We  have  found  (Art.  47)  that  the  differential  of  the  arc  of 
a  circle  is  equal  to 

R//sin. 


VR^-sin.2 
hence  we  have 
Xo=arc  of  a  circle  of  which  a  is  radius  and  x  is  the  sine. 

(200)  Integrate  the  expression 

/x^dx 
Va^-x^ 
Substitute  in  formula  a,  2  for  m  and  we  have 

'^    "        dx 


x^ 


«"    /»        ax  X   , 

2    J     V^2_  _^2  2 

which  is  equal  to 


X 


2        "2 

where  Xq  has  the  same  value  as  in  (Art.  199). 

Similarly  by  substituting  different  values  for  m  in  formula 
a  we  have 


INTEGRATION    BY    PARTS.  295 

/x'^dx  Ka^  x^      , 

Va'^—x^       6*6 

/X^dx  7^2  _^T        

V^2-^2  8         s  8 

in  which  the  values  of  X^,  Xo,  X^,  X^,  Xg  remain  the  same 
throughout.  Thus  formula  a  reduces  the  integral  of  a  dif- 
ferential of  the  form 

x'^dx 


V  a^—x^ 

to  that  of  one  depending  on  the  integrals  of  differentials  of 

the  forms. 

x'^-^^x  x^-\lx  x^-^dx 

and  so  on 


V^2_^2'        V^2_^^2'       V.Z--Jt:2 

until,  if  7n  is  an  even  number,  we  shall  after  'f  operations 
find  the  integral  of  the  given  differential  to  depend  on  that 
of  a  differential  of  the  form 

dx 

V^2_^2 

X 

which  is  the  differential  of  the  arc  of  a  circle  of  which  — 

a 

is  the  sine  (Art.  177). 

(201)  By  a  similar  substitution  in  formula  A  we  may  find 

FORMULA    b 

thus 

_  ,      x'^dx      _^^~^/-r-] — 7,     {in—\)a^  /»  x^-\ix 

X^n  = 


/,     x"'-dx  x""  ■"    / \i}i—\)a''  /»  x""' 

=  f—-===z Va'-i-x-'- ^—     —= 


If  then  we  have  the  expression 


1L 


r      x'^dx 


296  INTEGRAL    CALCULUS. 

we  would  make  /'//=4  in  formula  b  vvhich  would  then  become 
The  integral  of 


^    '  4  A  ^  Va^  +  x'' 


x'^dx 


V  a^-i-x^^ 
we  have  found  (Art.  195)  to  be 


X ,   ^         , ,  L      a 


{a^  -{-  x^)'^  —  — log.  {x-\-'\/a^  -hx^) 

hence 

/•    x^dx    _x^      ;^a^x     3<^*  

/      ,  — \/a^  +x2  —  —^ — \^a2  +X2  4-— -logri^  +  y  ^3  +x2) 

J   Vrt^  +x2         4,  o  h        ^^  ''  ' 

(202)  The  expression 


x^dx 
V  2ax —  x^ 


may  be  integrated  by  first  reducing  it  to  the  given  form  (Art. 
184)  and  making  the  proper  substitutions  in  formula  A  (iVrt. 

197)- 

It  may  however  be  integrated  by  an  independent  process 

as  follows  : 
Make 

^—^^m-i^  2^^^_^^2  =  (2ax^'^-^ — x^'^)^ 

and  we  have  by  differentiating 

a{  2in  —  I  ^y?''"^~^dx  —  ino^^~^dx 
dv^— 7 

{2ax^'''^-^  —  x^'^)^ 

which  becomes  by  dividing  the  terms  by  x^-'^ 

a{^2  7n  —  I )  x^^  ^dx         inx  '^dx 

dv=  7—  —  ■         J 

{lax — x'^y-^  {2ax — x^)^ 

Now  this  last  term  is  equal  to 

m  .  dX  m 

hence 

a{2m —  i)x'^~'^dx 
dv^— -7 —  — in  .  dXm 

{2ax — .v~)^ 


INTEGRATION    BY    PARTS.  297 

or  by  transposition 

m{2ax — jc^)^ 
and  by  integrating  and  substituting  the  value  of  v  we  have 

FORMULA    C 

p       x^dx  ai27n—\)r     x'"^~'^dx        x^-'^    , ^ 

/  ——:rr===— /   —  -— V   2aX—  X" 

^  V  2ax  —  3c2  m       J    V  2ax—  x^        ^^^ 

an  expression  which  depends  on  the  integral  of 

x"^-h/x 

V  2  ax —  x''^ 
in  whicn  the  exponent  of  the  variable  without  the  parenthe- 
sis is  diminished  by  one. 

(203)   If  we  take  the  expression 

dX^  — 


y  2ax — x'^ 

we  see  (Art.  47)  that  it  is  the  differential  of  an  arc  whose 
radius  is  a  and  whose  versed  sine  is  x  ;  or,  which  is  the  same 

thing,  an  arc  whose  versed  sine  is  —  and  radius  i  ;  hence 
°'  a 


a 

/adx  .         x 

,  =^=ver.  sin.~^— 

V  2ax—  x^  ^^ 

(204)  If  we  take  the  expression 

xdx 


^      '■    V  2ax—  x'^ 
and  make  m  in  formula  c  equal  to  i,  we  shall  have 

Xi  ~XQ  —  \/2ax—  x- 
in  which  Xq  has  the  same  value  as  in  (Art.  203). 

Similarly 

/x^dx  xa  X    / 

,  ,=^-Xt  -  -  V 2ax  -  x'^ 

V  2ax —  x~       ~  2 

and 


298  INTEGRAL    CALCULUS. 

C       x^dx  ^a  X^      , - 

^     ^  V2ax-x^       3      ^         3 
where  X^,  Xg,  X3,  have  the  same  value  throughout. 
Thus  formula  c  reduces  the  binomial  differential. 

X  '^dx 


V  2ax  —  x''^ 
to  depend  successively  on  the  integrals  of 

x^-\ix  x'^-hix  x^''-^dx 


V  2ax— x^^     V  2ax— x'^^     V  2ax— x^ 
and  finally  on 

dx 


V  'lax —  X" 
which  represents  the  differential  of  an  arc  whose  versed  sine 


X 

is  —  as  we  have  seen. 

a 


(205)  To  find  the  integral  of 


3. 

x'^dx 


y  2ax — x^ 
we  substitute  in  formula  c^  f  for  m  which  gives 

/x^dx  Aa  f*       x^dx  2X^      , — 

—  zi= —  /  — — -^;—  V  2ax—  X' 


Dividing  the  terms  of  the  first  fraction  in  the  second  mem- 

JL 

ber  of  the  equation  by  x^  we  have 

x^dx  dx 


V  2ax— x'^      V2a—x 
of  which  the  integral  is 

~^V2a—x 
hence 


/ 


x^dx  Sa    , 2X 


.  =  — V2^  — .T— V  2a  — ^ 

y  2ax—  x^  3  3 


INTEGRATION    BY    PARTS.  299 

(206)  The  method  of  diminishing  the  exponent  of  the 
variable  without  the  parenthesis  by  means  of  formula  A, 
will  of  course  only  apply  when  ;//  is  positive.  But  we  may 
obtain  from  this  another  formula  which  will  diminish  the 
exponent  when  it  is  negative.  To  do  this  we  multiply  the 
formula  A  by  the  denominator  and  we  have 

•or 

r  fx^-^  {a-\-bx^)P  dx  — 

J      xrn--n^\a-\-bx'^y'^  —  b{^np-\-m-\-\)fx'^{a^rbx'^Ydx 
(  a{jn  —  n-\-i) 

Making  m—n^=-  —  m  we  have 

Formula  B 

(  fx-'^{a + bx  ^)vdx—  \ 

)  x-'^^\a-]rbx^y^^-b{np-m-\-n-\-i)fx-^''^^{a^bx^ydxK 
(  ai^  —  m-\-\)  ) 

If  tn  denote  the  greatest  multiple  of  n  contained  in  }?i  we 
shall  have  after  /+ 1  reductions  the  integral  of 

x-'^ia-^bx'^ydx 
to  depend  on  that  of 

x-rn^^^^^^'^a-^-bx'^Y  dx 

and  if  —m-\-{t-\-\)n—n—\  we  shall  have  (Art.  164) 

{a-[-bx^Y^^ 

but  in  this  case 

—  m-\-\ 

n 
.a  whole  number;  and  hence  the  original  expression  maybe 
integrated  as  in  (Art.  186). 
(207)  To  find  the  integral  of 


300  INTEGRAL    CALCULUS. 

dx 


Substitute  In  formula  B 

2  for  m 
I  for  a 
I  for  b 

3  for  n 
-\  for  p 

and  we  have 

since  ^z(  — ;;/  + 1 )  =  —  i   and  b{^np  —  m  +;/  + 1 )  =  i . 

(208)  To  find  the  integral  of 

dx  -'^ 

x~'^(2 — x^)    ^dx 


3 

Substitute  in  formula  B 

2    for   7/1 

2  for  a 

—  I  for  b 

2  for  n 

-I  for/ 

which  gives 

3  13 

Jx~'^{2  —  x~)    ^dx=:—x~''^{2~x^)    ^  — 2/(2— x^y^dx 
since  a[^  —  ;;?  +  i)^  — 2   and  b{jip  —  in-\-n-\-\)'^=^2. 

(209)  Besides  the  method  of  reducing  the  exponent  of 
the  variable  without  the  parenthesis,  we  may  make  the 
integral  to  depend  on  that  of  another  expression  of  the  same 
form  in  which  the  exponent  of  the  parenthesis  itself  is  re- 
duced by  one.  This  is  the  second  general  method  referred 
to  in  (Art.  197). 

Let  us  make  v^=^x^  where  s  is  an  exponent  to  which  we 
may  assign  any  required  value.     From  this  we  obtain 

dv=^sx'^~''^dx  ([) 


Intel; RATION  by  parts.  301 

If  now  we  assume 

udv—x'^[a-\-bx''Ydx  (2) 

we  shall  have  by  dividing  equation  (2)  by  equation  (i) 

^Vfl  —  S^  1 

and 

;//  -_ J-  + 1  blip  ^ ,  .       ^         ,  , 

du  = ^ x'''-'{a-{-bx''ydx-h~^x^-'^\a-i-bx'')P-^x"-h^x 

but 

(a+bx'')P=(a-\-bx>'){a-\-l>x"y-^ 

hence 

a(7n—s-\-i)-\-d{m—s-\-i-{-/ip)x"'  ,      x      -,  , 

du  =  -^^ ^       ^ ~ — x^-'  {a-\-bx^  )P-h/x 

s 

Let  the  value  of  s  be  taken  such  that 

;;/ — s-\-i  +///=o 

or 

i"=///+ 1  -\-/^p 

and  we  shall  have 

—anpx'^-'(a-\-/Kx'')P-h/x 


du —  ,   ,        I 

///  -\-in-\-\ 

Substituting  these  values  of  u^  v^  du,  dv  in  formula  (i),  (Art. 

197),  we  have 

Formula  C 
^m^ i(,?+/;.r»)^'  -\-atipfxHa-\-hx 'M^^- V.r 

fx'^ia^-bx^Ydx  = ^ — ^_r   , 

in  which  the  integral  of  the  expression  is  made  to  depend 
on  that  of  one  of  the  same  form  in  which  the  exponent  of 
the  parenthesis  is  one  less  than  that  given. 

By  a  similar  process  this  last  may  be  made  to  depend  on 
the  integral  of  one  whose  exponent  of  the  parenthesis  is 
again  one  less;  and  so  on  until  the  exponent  of  the  paren- 
thesis shall  have  become  less  than  one. 


202  INTEGRAL    CALCULUS 

(210)  To  integrate  the  expression 

substitute  in  formula  C 

o  for  m 
a^  for  a 
\  iox  b 
2  for  n 
ifor/ 
and  we  obtain 

r,     / x^/ a'^j^x^      a^   r       dx 

fdx^a^^x^^ ^+VJv:^^ 

but  by  (Art.  194)  we  have  found 
dx 


/ax  


hence 


fdx\/  a^j^x^= ~ — -  +^log-(^  +  V«2_f_^3) 


in  like  manner  we  find 


X\/  -V* /J"        /x*  /  ^^^^^— — 

fdx^ x^-a^— — -log.  (^+V^2-a2) 


2  2 


(211)  If  the  exponent  of  the  parenthesis  is  negative,  this 
formula  will,  of  course,  not  answer,  but  we  can  easily  deduce 
from  it  one  that  will  effect  the  object.  For  this  purpose  we 
clear  it  from  fractions,  transfer  the  integral  term,  and  divide 
by  the  coefficient  of  the  last  term  in  formula  C  and  we  have 

Formula   D 

(  Jx'^{a-^bx'^y-^dx 

\  _-x'^^'^{a+bx")P-\-{np  +  7n-\-\)Jx'^(a-{-bx^)Pdx 
(  a/ip 

To  find  the  integral  of 

(2— :r~)    ^dx 


INTEGRATION    BY    PARTS.  303 

substitute  in  formula  D 

o  for  ?;i 

2  iox  a 

—  I  for  <^ 

2  for  n 

— I  for/- 1 

and  we  have 


^V  2-X^ 

To  find  the  integral  of 

xdx  _i. 

■l^=x{i-^x^)    ^dx 

(i+Jt:3)3 

we  substitute  in  the  formula 

I  for  7)1 

I  for  a 

I  for  b 

3  for  n 
—4  for /—I 
and  obtain 

fx(Y-\-x'^y^dx=-^{i^-x-^Y^-2fx{i^x^ydx 

2 

in  which  x{i-{-x^)^dx  may  be  developed  into  a  series,  and 
each  term  integrated  separately. 

INTEGRATION    OF    EXPONENTIAL    DIFFERENTIALS. 

(212)    It  has  been  shown  (Art.  37),  that  in  the  Naperian 
system,  the  differential  of  a^=a^  log.  adxy  whence 

/a^dx= , 

log.  X 

By  means  of  this   equation  we    may  integrate   the   general 
expression  a^Xdx,  in  which  X   is  a  function  of  x.     For  this 


304  INTEGRAL    CALCULUS. 

purpose  we  put   the  expression  into  the  form  y^.a^dx  ;  and 
applying  the  formula  (Art.  197) 

J  udv = 2rd  —fvdu 

by  making  u^^.  a^dx--dv,  and ^=v  we  have 

log.  a 

fX.  a^dx=^~-  fj^^dX  (i) 

log.  a      J  log.  a  ^  ' 

By  differentiating  successively  the  function  X  and  repre- 
senting the  successive  differential  coefficients  by  X',  X", 
X!"  .     .     .     .     X^'  we  shall  have 

dX^X'dx,  dX'=X"dx,  dYJ'-X"'dx,  etc., 
whence  we  have 

^    a^      ^^         r    X'      ^  ^  X'        ^      C       a^ 

/  7 ^^  o^  ./  7 d^dx^j-j xi"^       /  77 Vi-^X 

•^  log.  a  ^  log.  a  V<^S-  ^)  V^.^-  ^) 

and  substituting  this  value  for  the  last  term  in  equation  ^i) 

X  d"^         X'  d^  i*       d"^ 

we  \-\d.VQfX.d^dx—-r^ — — ^ — ^^-\-     -r-. vr.dX' 

log.  a      {log.  a)"     tl    [log.  a)"^ 

and  continuing  thus  to  operate  the  following  series  will  be 
developed 

X     -  x\     X     _       X^  X"      _      X'" 

^      ^  (  log.  a      {log.  aY      {/og.    «)3      {Jog.  d)^' 


^{log.aY^^  )  '^J  [log.  aY^'^ 


If,  in  finding  the  successive  differential  coefficients  X', 
X",  X'''.  .  .  .  X"-',  the  last  one  of  them  is  constant  we 
shall  have  ^X"''=o  and  of  course  that  term  of  the  integral 
vanishes. 

{I?)  Let  us  take  for  example  X=x^  then  we  have 
X'=3X^  X"  =  2  .  sx,  X'"  or  X'^'  =  2  .  3 
and 

fx''d''dx=a4    "^^^     -      ^^^      4-     ""'^"^     -        ^ 
^  ^    ^log.  a      {log.  a)"      {log.  a)^      {log.  a)'^ 

We  may,  also,  obtain  another  development  of  the  func- 


INTEGRATION    BY    PARTS.  305 

tion  a^Xdx  by  making  a^  —  n  and   ^dx^=^dv  in  the  formula 

already  used.     Then  representing 

fY.dx  by  P,  f^dx  by  Q,  f<^dx  by  R,  etc., 

and  integrating  by  parts  we  shall  have  iox  Judv^^uv—Jvdu 

I a^y.dx—a'''^ -J a^  log.  a  Vdx  (2) 

and 

Ja^  log.  a  Vdx=^a^  log.  a  Q,—Jd^  {log.  aYQ^dx 
and  substituting  this  value  in  equation  (2)  we  have 

Ja'^Xdx=a''V-a''  log.  a  Q,+JiT^  {log.  aYQ^dx 
and  continuing  thus  to  integrate  by  parts  we  have  in  gen- 
eral, 
Ja^Xdx—ax(V-(:i  log.  a-\-K  {log.  a)^  — ,  etc.) ±  J  Za^  {log.  ayhix 

\i  we  apply  this  formula  to  the  case  where  X=— .  we  shall 
find 

P  = L_    0=       '         R^ \ z=        ^ 


4X'^-    -     3  •4'-'^"'  2.z-4^^  2.3.4.T 

whence 

rc^dx_  x\  _    ^    _  ^^^'  ^  __  (^^g"-  ^)^    \      {log^_ci)^  fa'^dx 
J    x^  \      4X^     3 . 4 .  cr^      2  .  3  . 4  A'"  )        2.3.4'/      X 

The  integral  of can   not  be   determined   exactly  by 

any  method  yet  discovered. 

In  general  we   see  that  whenever  tne  exponent  of  x  is 
integral   and   negative,  the  integral  will   finally  depend  on 

CV^dx 

tluit  of  — — ■;  for  in  the  successive  functions  P.  O.  R.  etc., 
X  ^  ' 

the  exponents  of  x  diminish  constantly  by  one  ;  and  hence 

the  last  function  will  be 

/Aa^dx  _      ra^dx 
X  J      X 

A    being    any    constant   coefficient.       We   may    obtain    an 

approximate  value  of  the  integral  of  ^ by  substituting 


3o6  INTEGRAL    CALCULUS. 

in  the  expression  the  development  of  a^.     If,  in  Art.  37,  we 
make  v=^x  and  k^=^Iog.  a  we  find 

a^=^i-{-x  log.  a-\ •  [log.  aY -\ {log.  a)^-{-.,Qtc., 

and  substituting  this  value  of  a^  in  the  place  of  that  quan- 

tity  in  the  expression we  have 

(z^ilx      lix  X  X 

= Vlog.  a  dx-\-~  (log.  aY(lx-\ {^og.  d)^dx-{-^  etc. 

Integrating  each  term  separately  we  have 

—log.  x+x  log.  a-i {log.  a)^-\ — -  (log.  a)^+,  etc. 

X  4  I" 

If  we  make  u=x'"  in  the  equation  — =^1  {log.u)  or  rather 

du^=-iid  {log.  ii)  we  shall  have 

dx^^^x^d  {log.  x^^ 
so  that  whenever  we  can  decompose  a  differential  into  two 
factors  of  which  one  shall  be  represented    by  x^  and  the- 
other  by  d  {log.  x'^)  the  integral  will  be  x^. 

If  we  make  the  constant  {a)  in  Eq.  (i)  equal  to  the  base 
of  the  Naperian  system  of  logarithms  then,  since  the  log- 
arithm of  the  base  is  one,  the  foregoing  results  will  assume 
a  simpler  form. 

In  the  following  examples  the  base  of  the  Naperian  sys- 
tem is  represented  by  e. 

Ex.  I.     Find  the  integral  of  xe^dx. 

Ans.     e^{x—\).. 

Ex.  2.     Find  the  integral  of  x'^e^dx. 

Ans.     e%x^  —  2X-\-2). 

Ex,  3.     Find  the  integral  of  x^e'^dx. 

Ans.     e^(x^  —  4X^-\-i2X^--24X-{'24). 

xdx 
Ex.  4.     Find  the  integral  of  — ^ 


INTEGRATION    BY    PARTS.  307 


Assume  u=^Xj  and  dv^^e  ^dx  which  give 
du=^dx,  and  v=^—e~^ 
Hence 

/xdx  _„      ,    pdx 


Integration  of  Logarithmic  Differentials. 

(213)     The  method  of  integration  by  parts  may  also  be 
applied  to  integrate  the  expression 

Xdx  (log.  xY 
for  if  we   represent    yidx   by  dv.,  {log.  xY   by    ti   and    the 
integral   of    Xdx  by   X',   we   shall   have   z^=X'  and 

X 

X' 

X 

So  that  whenever  the  integral  of  \dx  can  be  found,  that 
of  Xdx{/og.  xy^  will  depend  on  the  integral  of  a  similar 
expression  in  which  the  exponent  of  /og.  x  is  one  less  than 
before.  Hence  if  n  be  entire  and  positive,  the  exponent 
of  /og.  X  will  become  zero,  and  the  expression  on  which  the 
integral  will  depend  will  become  algebraic. 
For  example,  let  us.  take  X=a-"\  then 

,7}l  t- 1 


''^~^dx 
du  —  n  {log.  x)      ■ — ;   and  hence  (Art  197) 

r  X' 

fXdx  {log.  xY  =  X.'  {log.  xY-nj  —dx  {log.  xY'^      (3) 


rx^dx=-^^^^yj 

^  m-\-i 


and  substituting  these  values  in  equation  (3)  we  have 
fx^\log.  xYdx^-^;^^{log.  xY—;;^^fx^Vog-  ocY-^dx 

It  will  be  observed  that  the  term  in  the  second  mem- 
ber of  the  equation  to  be  integrated  is  of  the  same  form  as 
the  original  quantity,  with  n—\  instead  of  n  as  the  expo- 


3o8  INTEGRAL    CALCULUS. 

nent  of  the  log.  xj  hence  by  substituting  n—i  in  place  of 
ji  we  have 

yx-(%.  x)^^dx  =  ^^^{log.  x)'^^--^^/x-^{log.  xr-^dx 
and  substituting  11—2  for  11— \  in  this  equation  we  obtain 


^m+l 


Jx-\log.  x)^^dx=-^^^{log.  xY-^-^^^fx-\log.  xY-Mx 

Continuing  thus  we  shall  finally  arrive   at  an  expression  in 
which,  if  n  be  integral  and  positive,  the  last  term  will  be 

\vi-\-\Y 

and  since  (log.  ^)^  =  i,  the  integral  of  x^{log.  x)^dx  will  be 


?n-\-\ 
and  the  term  becomes 

,  n{n—\\n  —  2){ii—'^  ....   2  .  i    „t+ 1 

.so  that  we  shall  have 

x"^^^  j  ,           ^       n(log.  x)'^'^ 
/x'^log.  x'^dx=^—, —  1  (log.  x^— — ; 

7t{7i—i){log.xY--'^  ;^(;^-I)(;^-2)(;^-3).  ■  ..2.1  )  ,   . 

{771+lY  ""-  {77l-\-lY  ^ 

It  will  be  observed  that  the  odd  terms  are  plus  and  the 
even  ones  minus,  so  that  if  71  be  an  even  number,  the  last 
term  will  be  an  odd  one  and  plus ;  if  ;^  be  an  odd  number,  the 
last  term  will  be  even  an.d  minus. 

If  we  make  /2  =  i  we  have  from  equation  (4) 


^m +1  J  ^m + 1 


fx'^log.  xdx= — 7—^^^-  ^ r-  •  r- 


lo§[.  X 

77t-\-\ 

771 -\- 


-  \  log.   X -j—  f 


INTEGRATION    OF    PARTS.  309 

If  we  make  w=4  and  n~i  we  have 
If  we  make  ;2  =  2  we  have 
If  we  make  vi^=-2  and  ;/  =  2  we  have 

/X^{l0g.  xYiix  =  ~^   \    {log.   xY  —  '^log.    X-\-%    r 

If  we  make  w=5  and  /^=3  we  have 

x^    i  //         \q_(/^^.  ^)^  ,   log  X      ^    ) 

If  we  make  ?/i  =  i  and  ;^  =  i  we  hrave 

x'^ 
/x  log.  xdx——  {log.  x  —  \) 

If  ;;2=o  and  ;2=i  we  have 

J  log.  xdx^x^log.  x—\) 
If  jn  —  —  \  the   second    member  of  the  equation   (4)   be- 
comes infinite.     In  this  case  the  first  member  becomes 


{log.  xY 


(ix 


If  we  make  log.  x=z  we  have   ^=ds  and 

.s^^+i     {log.  xY'-'^ 

/{log.  .Y  ?=Av^-,7+-r  --""^ 

which  is  true   for  all    values  of  //   except   n  =  —  i.      In   this 

case  the  expression  becomes — 7^ — .     If  we  make  log.  x=z 
^  X  log.  X 

we  have  -^  =dz  and 

f—r^—=  f^^l^K-  ^'^log.{log.  x) 
J  X  log.  x     *'     z 

If  ji  is  preceded  by  the  minus  sign   the   method  given  in 
equation  (4)  fails  because  the  exponent  of  the  log.  x,  instead 


3IO  INTEGRA;.    CALCULUS. 

of  approaching  zero,  continually,  becomes   greater.     In  this 
case  we  may  put  the  proposed  differential  under  the  form 

x'^^\log.  x)-''  ~  or  jv"^  +  i(%.  .t)-V^^.  X 

assume  2^=^x'^'^^,  and  d-o^^[log.  x)~^d log.  x,  which  give  du=- 

(m-\-\)x^dx  and  v=- — ^'  "  , and  we  shall  have 

—  n-\-\ 

-^         ^  (;/— 1)(%.  x)^^      n—iJ  {log.  xy^-^  ^^' 

If  n  be  a  whole  number,  the  repeated   application  of  this 
formula  will  at  length  give  for  the  term  to  be  integrated 

x^^dx 
log.  X 

In  order  to  simplify  this  expression  put 

los".  z 


whence  (;;^-|-i)  log.  x^^log.  5,  or  log.  x- 


m-\-\ 


Also,  by  differentiating. 

( ;;/  + 1 )  x'^'^dx  =  dz 

whence 

Hence 

x^dx         dz 

log.  X       log.  z 

a  differential  which  can  be  integrated  by  a  series. 
If  we  make  m^=-Af  and  n^=  —  2  we  shall  have 

/x^dx     _         -^^       I  ^  ^  x^dx 
(log.  x)'^         log-  X         J  log.  X 

If  we  make  7n=^4  and  ;/  =  —  3  we  shall  have 

/x^dx  x^  ^x^        25    rx^dx 

{log.  x)'^  ^{log.  x)"      2  log.  X      2  J  log.  X 


SECTION    III. 


(2  1 4)  Application  of  the  Integral  Calculus  to  the  Measure^ 
•meni  of  Geometrical  Magnitudes. 

We  have  seen  (Art.  173)  that  wlieii  two  differentials  are 
equal,  their  integrals  will  also  be  ec^ual  or  else  have  a  con- 
stant difference.  It  is  upon  this  principle  that  the  method 
of  measuring  geometrical  magnitudes  by  means  of  the  cal- 
culus is  founded.  We  obtain  the  expression  for  the  rate  of 
change  in  the  magnitude,  in  a  function  of  one  variable  and 
its  differential.  It  will  follow  that  the  magnitude  itself  is 
equal  to  the  integral  of  the  function,  or  else  the  difference 
between  them  will  be  constant  for  all  values  of  the  variable. 

Thus  let  M  represent  any  magnitude,  and  let  F(.v)^/.v  rep- 
resent its  rate  of  increase  while  being  generated  by  its  ele- 
ment—  that  is,  its  differential :  F(jc)  being  the  differential 
coefficient,  and  a  function  of  x. 

Then  we  have 

dW=Y{ix)dx 

Avhich  is  an  equation  between  two  differentials,  hence  the 
integrals  are  equal  or  else  differ  by  a  constant  quantity.  If 
we  represent  the  integral  of  Y[x\dx  by  X  we  shall  have 

M=X  +  C^ 
where  C   renresents   the  constant   difference    between    the 
quantities  whose  rates  of  change  are  equal. 

The  method  of  disposing  of  the  term  C  is  shown  in  (Art. 
173),  and  the  result  will  be  an  expression  for  the  value  of 


312  MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES, 

M  in  terms  of  one  variable.  Then  assigning  to  this  variable 
any  specific  value,  we  obtain  the  value  of  M  from  the  be- 
ginning up  to  that  value  of  the  variable;  or,  by  giving  to 
the  variable  two  successive  values,  tiie  difference  of  the  two 
resulting  expressions  will  give  the  value  of  that  portion  of 
M  lying  between  the  two  values  of  the  variable. 

RECTIFICATION    OF    CURVES. 

(215)  To  rectify  a  curve  is  to  find  what  would  be  its 
length  if  it  were  developed  into  a  straight  line ;  in  other 
words,  to  find  the  measure  of  its  length.  When  its  differ- 
ential can  be  obtained  in  an  integrable  form  it  is  said  to  be 
rectifiable. 

The  general  expression  for  the  differential  of  any  plane 
curve  whose  equation  is  referred  to  rectangular  axes  is 
(Art.  34)  

and  hence 

is  the  general  expression  for  an  indefinite  portion  of  any 
such  curve.  In  order  to  obtain  the  integral  of  this  expres- 
sion, we  must  know  the  relation  between  x  and  y  which  we 
obtain  from  the  equation  of  the  curve;  and  by  means  of  it 
eliminate  one  of  the  variables  and  its  differential  from  the 
formula ;  thus  producing  a  differential  function  involving 
but  one  variable  and  its  differential,  whose  integral,  when  it 
can  be  obtained,  will  be  the  length  of  an  indefinite  portion 
of  the  curve. 

(216)  To  find  the  length  of  a  Circular  Arc. 

We  have  in  (Art.  47)  several  expressions  for  the  differential 
of  an  arc  of  a  circle,  in  terms  of  its  trigonometrical  func- 
tions, which  already  contain  but  one  variable  in  each  case. 


INTEG'RAL    CALCULUS. 


If  we  select  that  in  which  the  tangent  is  the  variable,  we 
will  represent  it  by  /  and  the  formula  becomes 


(It  I 

(III = ' — -r~r  =  — ri^^ii 


Developing  the  fraction  we  have 
I 


hence 


,  ^1-/2 +/-!-/«+/« -etc. 

I  "T"^ 


,lu = — \—dt=dt—  /2  ^//+ /^  df—  /f'  dt-\-  etc. 


and  integrating  each  term  separately  wc  have 

/^        t^       O       t^ 

fdu^u=^t— — -+ — — 1 —  etc. 

-^  3        5         7         9 

or 

t-       i^       t^       t^ 

But  we  have  found  (Art.  i8o)  C=o,  and  it  we  assume 

2^  =  30° 
we  shall  have  _ 

and  substituting  this  value  for  /  we  have 

/— /  I  I  I  I 

u—VU\— +-  - — T— ^+ ::— etc. 

"^^       3  -3     5  ■  y     7-3'     9-3' 

which  being  reduced  is  equal  to  0.523598  nearly,  for  the 
length  of  an  arc  of  30° ;  and  multiplying  by  6  we  have  the 
arc  of  a  semi-circle  e(iual  to  3.141588  when  radius  is  i. 
Hence  this  is  the  ratio  between  the  diameter  and  the  entire 
circumference. 

(217)    To  find  the  length  of  an  Arc  of  a  Parabola. 

We  have  found  (i\rt.  57)  that  the  differential  of  an  arc  of 
a  parabola  is 


314         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

The  integral  of  this  (Art.  210)  is 

If  we  estimate  from  the  vertex  of  the  curve  where  u^=-q,  and 
y=^o  we  shall  have 


hence 


<>=flog./+C 


C  =  -4log./ 


Substituting  this  value  of  C  we  have  for  the  definite  integral 

^^-^V^M^+^log.  (v+V7Mry^)-|log./ 

or 


U 


^irVZ-^+y+^io^ 


2p        ^  -^  2        ^  P 

(218)     To  find  the  length  of  an  Arc  of  an  Ellipse. 

We  have  found  (Art.  57)  that  the  differential  of  an  arc  of 
an  ellipse  is 

If  we  take  c  to  represent  the  distance  from  the  center  to  the 

focus  of  the  ellipse  we  have 

B23=A--^2 

hence 


du  ^—dxsj  -— — 

i\        V      h.'^—x" 

If  now  we  represent  the  eccentricity  of  the  ellipse  by  e  we 

have  r=A<?,  and  hence 

T     ,       /A^-A=^^'^- 


du=^—7-dx 


V 


A-V       A'-x^ 


INTEGRAL    CALCULUS.  315 

or,  dividing  by  A"*   under  the  radical  and  multiplyinf;;  by  A^ 
without  it  we  have 

2~ 


du = k.dx  v  


h}-x 

Developing  (i  — -r:j  )     l^Y  the  binomial  theorem  we  have 


1 

e- X' 


yi—~r^]    =1— —r^— TT— ■? — 7-^— etc. 


1 
e-'x-'Y  <^'^^'"  e*x*  y^x 

T^/  '^^~^aJ~2  .4.  a4~2  .4.6.  a 

hence 

Adx  c^         x'^iix  <'"*  x'^dx 

die 


.  VA--X^        2  a*    V  A~-^V~         2.4A3  •    V  A2_^2 

3<?6  x^dx 

"^.4-6A5  •   Va^'^^"^^'"' 
Making 

Adx  x^dx  x^dx 

—  ~  =^  X  r. , — ,  -=:^/X.T,  — ,  •  -=d\.,    etc. 

we  have 

^2    ^  <?■*  3t'*^ 

A///  =  Xo  —  — rXo— ttX, —  .  .  -Xp,  —  etc.        (i) 

^  "     2  A    "     2  .  4A"^     *     2.4.  GA"^     ^  ^ 

Now  by  (Art.  199)  Xq=  the  arc  of  a  circle  of  which  A  is 

the  radius  and  x  the  sine,  and  (Art.  200) 

A  X    , 

Xo=-x„--Va=^-^3 

"22 

also 

3A^  X 


and 


X .  =- — X.  - — V  A~  -x^ 
44-4 


qA2  x^     , 

x,=-^x,-— Va^-a--^ 


6       -t        6 

If  we  make  .t=:o  and  estimate  from  the  extremity  of  the 
■conjugate  axis,  we  have  ?/=o  and  C=o.  If  we  make  ji'  =  A 
we  shall  have  ?<^=  a  quadrant,  and  since 


3l6         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

we  have 

A  sA^  3A3  3  .  cA^ 

2"        *         4  2.4"        *^      2.4.6" 

and  substituting  these  values  in  equation  (i)  we  have 

V   .         ^^  3^^  3-  3-  5^^  ^    X 

?^=Xo(i— — — 7 — >  — etc.) 

"^       2.2     2.2.4.4     2.2.4.4.0.0  ' 

for  one-fourth  of  the  circumference  of  an  ellipse;  X^  being 

one-fourth  of  the  circumference  of  a  circle   of  which   the 

diameter  is  equal  to  the  major  axis  of  the  ellipse. 

Hence  the  whole  circumference  is  equal  to 

(?-  Y'^  3  ■  3  ■  5^^ 

2-A(i  — — '■ — 7 — 7— etc. 

^       2.2     2.2.4.4     2.2.4.4.6.0 

It  will  be  seen  that  as  the  eccentricity  diminishes  the  cir- 
cumference of  the  ellipse  approaches  the  value  of  27rA, 
which  it  reaches  when  e=o,  and  the  curve  becomes  a  circle. 

(219)      To  find  the  length  of  the  Arc  of  a  Cycloid. 

We  have  found  (Art.  129)  that  the  differential  equation  of 
a  cycloid  is 


dx  = 


V  2ry~y~ 
By  substituting  this  value  of  dx  in  the  formula  we  have 

d2i^=^  V dx'^  -\-dy"  ^=dy\/ -\- 1  =^dy\/ • 

V   2ry—y''^  V   2ry — y'^ 

or 

_  _i.  _  1  

U^y'  2rJ'dy{2r—y)     ^  =  —  2\/ 2r{2r—y)^  =^  —  2^2r{2r  —}'-\-C 

If  we  estimate  the  arc  from   D'  (Fig.   54)  where  jF  =  2r  we 
shall  have  ?/=o  and  C=o,  and  hence  making  _y  =  FG 

u  =  T>'F  =  —  2V 2r  {2r—y)  (i) 

We  see  from  the  figure  that 

D'E'=2r  and  D'H.'  =  =^2r—y 
hence 


INTEGRAL    CALCULUS.  317 

V2ry2r-y)=^VD  E'  .  D'H' ^D'F' 
so  that   the  arc   of  a  cycloid  is  eciual  to   twice  the  corr^-- 
pondiiig  chord  of  the  generating  circle. 

If  we  take  the  arc  D'A,  the  corresponding  chord  of  the 
generating  circle  becomes  the  diameter  D'E',  and  half  the 
arc  of  the  cycloid  is  equal  lo  twice  the  diameter  of  the  gen- 
erating circle,  or  the  entire  arc  is  equal  to  four  times  that 
diameter.     Thus,  if  we  make/=o  we  have 

or 

D'FA  =  2D'E'   and  AD'B=4D'E' 

(220)    To  find  the  length  of  the  Arc  of  a  Logarithmic  Spiral. 

We  have  found  (Art.  77)  that  the  differential  of  an  arc  of 
a  polar  curve  is 

du  =  ^  r-dv~^dr~ 
and  the  equation  of  the  logarithmic  spiral  is 

z'~Log.  r 
Hence 

yidr  ^      MVr3 

dv^ and  dv''^=^ 


r~ 


Substituting  this  value  of  dv"^  we  have 


du~^\i^dr-~  -f-  dr~  —drV  M'  +1 
In  the  Naperian  system  ]\I  =  i  and 

dic^=^dr\/  2 
hence 

ii  =  r\~2-\-C 
If  we  estimate  the  arc  from  the  pole  where  r=o  we  shall 
have  C=o  and 

u--r\/  1 
That   is,  the  length    of  an    arc  of  a  Naperian   logarithmic 
spiral  estimated  from  the  pole  is  equal  to  the  diagonal  of  a 
square  of  which  the  radius  vector  is  the  side. 


3l8         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

(221)    To  find  the  length  of  an  Arc  of  the  Spiral  of  Archimedes^ 

The  equation  of  the  spiral  (Art.  84)  is 

r^=^av 

I 

m  which  ^  =  —  and  e'=  the  arc  of  the  measurins;  circle  whose 

radius  is  the  value  of  r  after  one  revolution.     Hence 


du^=V  r^  dv'^  -\-d/^  =advV  1  -\-2r 
the  integral  of  which  may  be  found  in  (Art.  210).     Substi- 
tuting I  for  a  and  v  for  x,  thus 


^Vi-hz'  ^ 


u 


f  ^'V  i-\-v  "  ,  ,  -,  \ 

=a{ ^ +jiog.(e;+Vi+z,'-^))+C 


Estimating  the  arc  from  the  pole  where  z^=o  we  shall  have 
C=o  and 

u—^\y-\r^  I  +z;  ^  +log.(z^+ V  i  -\-v  2)] 
(222)    To  find  the  length  of  an  Arc  of  a  Hyperbolic  SpiraL 


The  equation  of  this  spiral  is  (x\rt.  86) 

rv 
Differentiating  we  have 

whence 


ab 
rv^=-ab  or  r= — 

V 


abdv 
dr  —  —^-^ 


/a^b'-dv'^      a^b^  ,  .     'i^'^^'^'     /  i 

and  

u  ^=^abfv''  "dv\/  27~  -j- 1 

Integrating  by  formula  B  (Art.  206) ;  making  in  the  formula 

;/^  =  2 

b  =  i 
n=^2 


INTEGRAL    CALCULUS.  319 

we  have 


or  (Art.  221) 


=  _^;-i(,+,2)l+,[!^^i±^'+ilog.(r.+V,+..^)] 


hence 


u  =^^[-z;-i(i  ^-v"Y-\-v\/  I  +  z^-  +  log.  (z'  + V  1  +  z^')]  +C 

Estimating  the  arc  from  the  point  where  z'=o  we  have 

3. 

which  is  as  it  should  be,  since  from  the  equation  of  the 
curve,  when  v=^o  the  radius  vector  is  infinite.  As  7-  is  infi- 
nite when  r  =  o  we  shall  have  u^^o  at  the  same  time. 
Hence  the  curve  is  unlimited  in  but  one  direction.  \\'e  may, 
however,  find  the  length  of  any  intermediate  portion  by  sub- 
stituting the  two  corresponding  values  of  v  in  the  integral 
function  and  taking  the  difference  of  the  results. 

(223)  Quadrature  of  Curves. 

The  quadrature  of  a  curve  is  the  process  of  finding  the 
measure  of  a  plain  surface  bounded  wholly,  or  in  part,  b}  a 
curve. 

To  find  the  area  of  such  a  surface  we  must  find  its  dif- 
ferential in  a  function  of  one  variable,  which  being  integrated 
will  give  an  expression  for  an  indefinite  portion  of  the  area, 
from  which  any  specific  portion  may  be  obtained  by  assign- 
ing corresponding  values  to  the  variable. 

(224)  To  find  the  area  of  a  Semi-Parabola. 

We  have  (Art.  65)  for  the  differential  of  the  surface  of  a 

parabola  _    j^ 

d  S  ^ydx = V  Vpx  ^  dx 

of  which  the  integral  is 


320        MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES, 

_      3. 

But  when  x  =  o  we  have  S=o,  and  hence  C^o;  so  that 
the  surface  of  a  parabola  bounded  by  the  curve,  the  axis 
and  an  ordinate  is  'equal  to  two-thirds  of  the  rectangle 
described  on  the  ordinate  and  corresponding  abscissa, 

(225)  To  find  the  aj'ea  of  any  Parabola. 

The  general  equation  of  the  parabola  is 

y'^^  '=^ax 

from  which  we  obtain 

ny'^~'^dv 

dx^=^ — 

a 

hence 

^■^  J       a         yn  +  i)a     n-\-i  -^ 

If  we   estimate  the  curve  from   the  origin  where  S=o  we 
have  ^=o,  and  hence  C=o,  and 

wb  —  — , — xy 

That,  is,  the  area  of  that  portion  of  any  parabola,  bounded 
by  the  curve,  the  axis  and  the  ordinate,  is  equal  to  the  rec- 
tangle   described    upon    the    ordinate    and    corresponding 

abscissa,  multiplied  by  the  ratio       ,     ,     If  n~~-2^  as  m   the 
common  parabola,  we  have 

If  /2=f,  as  in  the  cubic  parabola,  we  have 

If  ;2  =  i,  the  figure  becomes  a  triangle  and  we  have 

or  half  the  base  into  the  height. 

(226)  To  find  the  area  of  a  Circle. 
We  have  (Art.  d^)  for  the  circle 

ydx ^^dxy/  j^s — x^ 


INTEGRAL    CALCULUS. 


321 


Making  R  =  i  we  have 

d  S  =;Y/a:  =c/x\/  i  —  jc*  =^/jic:(  i  — JC^  )^ 

Developing  the  binomial  and  multiplying  each  term  by  dx 

we  have 

x^dx     x^dx     x^dx     ^x^dx 

d^—dx—  — — ■  —  — r^  —  — —- 3— —  etc. 

2  8  It)  128 

from  which  we  obtain  by  integrating  each  term  separately 

i^v  u\^  fcA*  S  »^V 

S=ji:— -7-  —  —  — — ~~— —  etc.+C 

6        40        112      1152 

Estimating  the  area  from  the  center  where  jf— o  we  have 

.S=o,  and,  therefore,  C=o,  so  that  the  series  expresses  the 

area  of  any   segment   1)etween   the  ordinate  at  the   center 

where  x^o  and   the  ordinate  corresponding  to  any  other 

value  of  X.     Hence  if  we  make  .t  =  i  we  have  the  area  of  a 

•quadrant  equal  to 

which  by  taking  enough  terms  may  be  reduced  to 

•78539 
Hence  the  entire  area  of  the  circle  will  be  equal  to 

3-14156 
equal  to  ~  where  radius  is  i. 

(227)  We  may  also  find  the  area  of  a  circle  by  consider- 
ing it  as  being  described  by  the  revolution  of  the  radius 
about  the  center.  In  this  case  the  radius  of  the  circle  be- 
comes the  radius  vector  and  we  have  (Art.  82) 

2R 
where  v  represents  the  arc  of  the  measuring  circle  and  R  its 
radius.     Integrating  the   terms  of  this  equation   we  have, 
since  r  is  constant, 

o 

7'"  7' 

Estimating  the  area  from  the  beginning  where  S=o  we  have 
i:'  =  o,  and  hence  C  =  o,  and 


32  2         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES, 

is  the  measure  of  a  sector  of  a  circle  of  which  z;=the  meas- 
uring arc.  Making  R=r  we  shall  have  z^=the  arc  of  the 
given  circle,  and  equation  (i)  becomes 

rv 

2 

that  is,  the  measure  of  a  sector  of  a  circle  is  half  the  pro- 
duct of  the  radius  into  the  arc  of  the  sector ;  and  hence  for 
the  entire  circle,  the  area  is  equal  to  half  the  product  of  the 
radius  into  the  circumference. 

If  we  make  z^=  the  entire  circumference  we  have 

27=27rR 
and  substituting  this  value  in  equation  (i)  we  have 

2K 

that  is  the  area  of  a  circle  is  equal  to  the  square  of  the 
radius  multiplied  by  the  ratio  between  the  diameter  and  the. 
circumference. 

(228)  To  find  the  area  of  an  Ellipse. 

We  have  in  the  case  of  an  ellipse  (Art.  64) 

B 


B       / 

^S ^=ydx^^-rdxy  A^  —x^ 


hence 

^=j^f{h.^-x''Ydx 

Integrating  by  formula  C  (Art.  209),  and  substituting 

o  for  m 

A2  for  a 

—  I  for  b 

2  for  n 

ifor/ 

we  have 


INTEGRAL    CALCULUS.  3^Z 

1 

but  (Art.  178) 

/(A2-A-)    V.r^  /  — =:sin.-'^-r 

hence 

•      B       , AB  X 

2A  2  A 

Estimating  from  the  center  where  x=o  we  have  S=o,  and 
hence  C=o.     Making  then  jc=A  we  have 

AB  .  AB      - 

S— — sin,~ii= —  .  — 
2  22 

for  one-fourth  of  the  area  of  the  ellipse,  since  the  arc  whose 
sine  is  i  is  equal  to  one-fourth  of  the  whole  circumference ; 
and  we  have  for  the  area  of  the  entire  ellipse 

S=-AB 
We  may  also  observe  that  (Art.  6;^) 

dxV'AF^^^ 
is  the  differential  of  the  area  of  a  circle  whose  radius  is  A, 

B 

hence  the  area  of  an  ellipse  is  T  x  the  area  of  the  circum- 
scribing circle  which  is  ~A-  ;  and  is,  therefore,  equal  to 

B 

-  .  -A2  =:-AB 

If  A=B  the  expression  becomes 

-A2  or  rR2 
for  the  area  of  a  circle. 

(229)    To  find  the  area  of  a  Segment  of  a  Hyperbola, 

We  have  in  the  case  of  a  hyperbola 

B    , 

jj,  =  -V  jc2— A2 

whence 


324         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 


B 


Integrating  by  formula  C  (Art.  209)  we  have 

S=-r-^V^2_A3- log.  (X  +  Vx^-K^)+C 

2A  2  *=    ^  ^ 

To  find  the  value  of  C  we  make  x—A.  where  S=o,  and  we 
have 

0  =  -— log.A+C 

hence 

AB 


C=—  log.  A 
2       ^ 


and 


B  ../-:^-r-.    AB^_  z^^+V^T^-A' 


S=—-xVx^-A^- — log.  (- 
2A  2       *=  v 


2A  2        "=■  V  A  "^ 

-wliich  represents  a  portion  of  the  area  between  the  curve 
and  the  ordinate  lying  on  one  side  of  the  axis.  Hence  the 
.area  of  the  entire  segment  cut  off  by  the  double  ordinate  is 


but 


|^^^3_A^-ABlog.(^^^"^^^^'     ^') 
B    , 


A 

hence 

fx      v\  /A^+B^\ 

S=^J-AB  log.  [-^+^)=xy-A'B  log-  [     ^^     ) 

for  the  value  of  the  area. 

(230)  We  may  also  find  the  area  of  that  part  of  the  sur- 
face lying  between  the  curve  and  the  asymptotes,  by  using 
the  equation  of  the  hyperbola  referred  to  its  center  and 
asymptotes,  which  is 

xy=m 
but  as  the  asymptotes  are  not  usually  at  right  angles  to  each 
other,  Vv^e  must  introduce  into  the  expression  for  the  differ- 
.ential  of  this  area,  the  sz'ne  of  the  angle  which  they  make 


INTEGRAL    CALCULUS.  3^5 

with  each  other  (Art.  58)  which  we  will  call  v.     We  shall 
then  have 

^/S=sin.  v.yi/x=^sin.  v 

and 

S  =  sin.  V  .m.  \og.x-\-Q, 

If  we  call  the  abscissa  of  the  vertex  i,  and  estimate  front 
the  corresponding  ordinate,  we  shall  have  at  that  point 

?n^i,  S=:o,  log.  x=o  and  hence  C^o 
And  since  sin.  v  may  be  considered  as  the  modulus  of  a  sys- 
tem of  logarithms,  we  may  make 

S  =  M  .  log.  .r  =  Log.  X 
That  is,  the  area  between  the  curve  and  the  asymptote^ 
estimated  from  the  ordinate  of  the  vertex,  is  equal  to  the 
logarithm  of  the  abscissa,  taken  in  a  system  whose  modulus 
is  the  sine  of  the  angle  made  by  the  asymptotes  with  each 
other. 

(23 1 )  To  find  th:  area  of  a  Cycloid. 

We  have  (Art,  129) 

ydv 

^^=  w =i 

V  2ry—y^ 

hence 

y^  dy 
dS  =ydx 


V  2ry—y^ 
Integrating  this  by  formula  c  (Art.  202)  we  have 

S=|-^' .  ver.  sin~^y— — —V2ry—y^~\-C 

Estimating  the  integral  from  A  (Fig.  76)  where  y=o,  we 

have 

S=o  and  hence  C=o 

and  taking  the  integral  where 

)'  =  2r=DE 
we  have 

S  =  |r .  ver.  sin.~^2r=3 


326  MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

that  is,  the  area  ADE  is  equal  to  three  times  the  semi-circle 

DF'E.     Hence    the  entire  area  of  the    p      Q _^  P 

cycloid  is  equal  to  three  times  the  area 
of  the  generating  circle. 

(232)  Another  method  of  obtaining 
the  area  of  a  cycloid  is,  to  consider  that  "^ 
portion  of  the  rectangle  ACDE  which  Fig.  76. 

lies  outside  the  curve. 

If  we  make  GF  =  2r—y=v  we  shall  have  the  differential 
of  the  area  DCAF  equal  to  v^fx  or 

y^y  / r 

^/S'  =(2r—y)(/x  =  {2r''y)—T^==  =dyy  2ry—y'^ 

V  2ry—y'^ 

Now  if  we  take  the  equation  of  a  circle  with  the  origin  at 

the  extremity  of  the  diameter  we  shall  have 

ydx^^dxy  27' x — x^ 

which  is  the  differential  of  the  segment  of  a  circle  of  which 

X  is  the  abscissa.     Hence  dys/  2ry—y'^  is  the  differential  of 

a  circle  of  which  y  is  the  abscissa,,  that  is  of  the  segment 

r'BE.     Hence  the  two  areas  ACGF  and  F'BE  have  the 

same  rate  of  change  or  differential  for  the  same  value  of  y  ; 

and  since  they  are  both  equal  to  zero  when  j=o,  they  are 

equal  for  every  other  value  of  j  (Art.  173),  and,  of  course, 

when  y=-2r.     Hence 

ACD  =  DF'E 


2 
But  the  rectangle  ACDE=7rr  .  2r  =  2-r2,  hence 

ADE=ACDE-ACD  =  '^-^^ 

2 

as  we  found  in  (Art.  231). 

(233)    To  find  the  area  bounded  by  the  coordinate  axes  and 
the  logarithmic  curve. 

We  have  had  (Art.  137)  for  the  logarithmic  curve 

yidy 
jc=Log.  J  and  dx^=^ 


INTEGRAL     CALCULUS. 


327 


hence 

d'^=ydx=lsldy  and  S  =  Mj+C 
If  we  estimate  the  area  from  AD  (Fig.  56)  where  >'=i  we 
have 

o  =  M-rC 
whence 

C  =  -M 
and 

S  =  MO-i) 
If  we  make_>'=o  we  have 

S  =  -M=area  ADD' 
If  j  =  2=P'Twe  have 

5  =  M=area  ADP'T 
So  that  although  the  axis  of  abscissas  is  an  asymptote  (Art, 
88)  to  the  curve  on  the  negative  side,  and,  therefore,  will  not 
meet  it  within  a  finite  distance,  yet  the  area  er.closed  be- 
tween them  is  limited  and  equal  to  ADP'T, 

(234)  If  we  take  the  curve  represented  by  the  equation 


y  = 


X 


to  which  the  axes  of  coordinates  are  asymptotes,  we  shall 
find  a  case  somewhat  similar. 

Putting  the  equation  into  the  form 


x^- 


r 


we  have 


and 


hence 


dx^=  — 


2dy 

yZ 


2dy 
d  S  =^ydx^=-—^^ 


S=-+C 

y 


fig-   77 


328         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

If  we  estimate  the  area  from  the  line  AC  where ^j'^^  00  we 

have 

o=o  +  C 

hence 

2 
C=o  and  S=— 

y 

If  we  makeji'  =  i=FT  we  have 

S=:2:=ATDC 

that  is,  the  area  ATDC  is  equal  to  twice  the  square  AHFT^ 
and  is,  therefore,  finite,  although  the  curve  FD  does  not 
meet  the  axis  AC  at  a  finite  distance. 

If  we  take  the  area  between  the  limits  y=^i  andjt'^o,  we 
shall  have 

S=|— 2 
that  is,  the  area  FEBT  is  infinite,  although  AB  is  likewise  an 
asymptote  to  the  curve. 

(235)    To  find  the  area  described  by  the  radius  vector  of  the 
Spiral  of  Archimedes. 

The  differential  of  the  area  of  a  polar  curve  (Art.  82)  is 

r'^dv 

V  being  the  arc  of  the  measuring  circle  and  R  its  radius. 
The  equation  of  the  Spiral  of  Archimedes  (Art.  84)  is 

r^^av 
hence 

dr'=adv 
or 

dr 
a 
Hence,  making  R  =  i  we  have 

/r"d7^       Cr"dr      r^ 
— =/ — =7-+c 
2        J     2a       oa 

If  we  make  r=o  we  have 
21 


INTEGRAL    CALCULUS.  329 

S=o  and  hence  C=o 

r 

and  since  a^—  we  have 

V 

Making  27=2-  we  have  for  the  area  described  by  one  revo- 
lution of  the  radius  vector 

3 
that  is,  the  area  described  by  one  revolution  of  the  radius 
vector  is  one-third  of  the  area  of  the  circle  described  with  a 
radius  equal  to  the  last  value  of  the  radius  vector. 

If  the  radius  vector  make  two  revolutions  we  have  v=/^ 

and 

2-r  " 


3 
where  r  =2/'  and 

S= — 

But  in  making  two  revolutions,  the  radius  vector  describes 
the  first  part  of  the  area  twice.  This,  therefore,  must  be 
subtracted,  and  we  have  the  area  enclosed  by  the  curve  and 
radius  vector  after  two  revolutions  equal  to 

^..r        3'^r        3'^/- 
and  by  subtracting  the  first  again  we  have  the  increased  area 
described  during  the  second  revolution  equal  to 

3/cr         ■^I'-r         ^'./ 

After  in  revolutions  we  have 

_  '  '> 

s= 


where  /  =^??ir,  hence 

m~ni  ■'?■■'      nrnr"  ,  . 

S= =^ (i) 

Subtracting  from  this  the  area  described  by  the  radius  vec- 
tor during  {jn—Y)  revolutions  we  have 


330         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

S'  =  — ;-(w^  — (;« — i)^)  (2) 

3 

Substituting  {m-{-i)  in  place  of  7m  we  have 

S''='^({m-i-iy-m^)  (3) 

for  the  area  described  by  the  radius  vector  during  the 
{m-i-i)//i  revolution.  Taking  the  difference  between  equa- 
tions (2)  and  (3)  we  have  the  additional  area  described  by 
the  {m-\-\)th  revolution  of  the  radius  vector,  that  is  the  area 
lying  between  the  mth  and  {^m-\-\)th  spires  thus 

S"  —  S'  = ((;;/  + 1 )  ^  —  2m^  -\-  {jn —  i  )^)^^2?n7:r^ 

We  have  found  the  additional  area  described  by  the  radius 
vector  during  the  second  revolution  equal  to  27rr^,  h'^nce  the 
additional  area  described  during  the  {j?i-\-i)th  revolution  is 
equal  to  m  times  that  described  by  the  radius  vector  during 
the  second  revolution.  That  is,  the  increase  of  the  additional 
areas  described  by  the  radius  vector  during  successive  revo- 
lutions, is  uniform  and  equal  to  twice  the  area  of  the  circle 
described  with  a  radius  equal  to  the  radius  vector  after  one 
revolution. 

If  the  area  ABP  (Fig.  34)  be  required,  that  is,  the  addi- 
tional area  corresponding  to  the  arc  BC  described  after  the 
first  revolution,  we  shall  have 

2~ 

z;=27r-j-- — ■ 
n 

and 

r  — r-\- 
n 

and  the  required  area  will  be 

^  n  r\^      TTr"* 

x     ^       n'  7, 


or 


s= 


INTEGRAL     CALCULUS.  33 1 

r(//+i)       r2(;/+i)2       -r^ 
2,fi  n^  3 


;f(^^+i)^ 


3^' 
Developing  (/z+i)^  we  have 

_^2 


3/^-  ^      -      "      '     3 

lOr 

s= — (1+-+-^+— )- — = — (i+-+ — -] 

3    "^         n     ft"       n^  f         3  )i    ^         n      yf  f 

2~ 

If  BCD  =1  circumference  = — ,  then /^  =4  and 

4 

(236)  To  find  the  area  of  the  surface  described  by  the  radius 
■vector  of  the  Hyperbolic  Spiral. 

The  equation  of  the  hyperbolic  spiral  (Art.  %(i)  is 

ab 
rv=-ab  or  r== — 

V 

in  which  a  is  the  radius  of  the  measuring  circle  and  b  is  the 
unit  of  the  measuring  arc.     Hence 

rr^dv      c  ab'dv  ab^ 

s=/ — =   — 2-=- — 

J      la  J       2V^  2V 

which  is  infinite  when  7'=o,  and  zero  when  z'=  00.     If  we 
make  v=-b=-AB  (Fig.  t,^),  and  7'=^<^=Aj-,  we  shall  have 

ab      ab      PB  .  OR 

^^ab—  — — — = 

22  2 

(237)  To  find  the  area  described  by  the  radius  vector  of  a 
Logarithmic  Spiral. 

We  have  (Art.  87)  for  this  spiral 

z/=Los.  r 


332  MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

and 

Mdr 

^v= 

r 

Substltuting  this  value  in  the  formula,  and  making  R  =  i  and. 

M  =  i,  we  have 

If  we  estimate  from  the  pole  where  S=o  we  have  r=o,  and 
hence  C=o,  and 

S=— 

4 

That  is,  the  area  described  by  the  radius  vector  of  the  Nape- 
aian  logarithmic  spiral  is  equal  to  one-fourth  of  the  square 
described  upon  the  last  value  of  the  radius  vector. 

(238)  Areas  of  Surfaces  of  Revolution. 

We  have  seen  (Art.  66)  that  the  area  of  a  surface 
of  revolution,  where  the  axis  of  revolution  is  the  axis  of 
abscissas,  is 

the  radical  part  being  the  differential  of  an  arc  of  the  re- 
volving curve. 

To  apply  this  formula  to  a  particular  case  we  must  obtain 
from  the  equation  of  the  revolving  curve,  the  value  of  one- 
variable  and  its  differential  in  terms  of  the  other,  so  that, 
when  substituted  in  the  formula  we  may  have  the  differen- 
tial of  the  surface  in  terms  of  one  variable  which  can  then 
be  integrated. 

(239)  To  find  the  convex  surface  of  a  Cone. 

We  have  (Art.  67)  for  the  differential  of  the  convex  sur- 
face of  a  cone 

<■/ S  =  2 ~axdx^J  ^3  _|_  J 

in  which  x  is  the  length  of  the  axis  and  a  is  the  tangent  ot 


INTEGRAL    CALCULUS.  333 

the  angle  made  by  the  revolving  element  of  the  cone  with 
the  axis  of  revolution.      Integrating  we  have 

Estimating  from  the  vertex  where  S=o  we  have  x=o,  and 
hence  C=o  and 

But  from  the  equation  of  the  generating  line  we  have 

y^ax 
and  hence 


and  by  substitution  we  have 


X' 


S~yxsy^^^=::yVx'^-j-/^ 

•or  (Fig.  T,i)  making  a:=AB 

S=-CB'^AB2+LB3 
that  is,  the  convex  surface  of  a  cone  is  equal  to  the  circum- 
ference of  the  base  multiplied  by  half  the  slant  height. 

(240)  For  the  convex  surface  of  a  cylinder  we  have 

j'  =  R ^radius  of  the  base 
Tience 

S  =/27zyV\/x'^-i-ay  =/2-Rc/x=2-Rx 
that  is,  the  convex  surface  of  a  cylinder  is  equal  to  the  cir- 
cumference of  its  base  into  its  altitude 

(241)  In  the  case  of  the  sphere  we  have  (Art.  6S) 

hence 

S  =  2-R.v  +  C 

Estimating  from  the  center  where  x=o  we  have  S=o  and 

hence  C=o,  and  the  measure  of  an  indefinite  portion  of  the 

convex  surface  of  a  sphere  is 

S  =  2rR.r 

the  same  as  that  of  the  circumscribing  cylinder  having  the 

same  altitude. 


334         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

Making  ji;=R  we  have 

S  =  2-R2 
for  the  measure  of  the  convex  surface  of  half  the  sphere  ; 
hence  for  the  entire  sphere  we  have 

S=4-R8 
or  four  great  circles. 

(242)    To  find  the  surface  of  an  Ellipsoid  described  by  anr 
ellipse  revolving  about  its  major  axis. 

Making 

^/  dx^^-dy^^du 
we  have 

2Tiy\^ dx^  -\-dy^  ^=2'!zydu 
But  we  have  found  (Art.  218) 


Kdx       f       cx"        e'^x'^  3<?"jr" 

jy/A2_j^2V^~2A2  ~2  .   aA*~2   .  A  .    6A6~^^W 


e'^x^ 

e^x^ 

y^x^ 

2A2 

2  .  4A* 

2 .4. 6A6     ^ 

e^x^ 

e^x^ 

y^x^ 

du  =  — 7^ 

hence 

2~A.vdx  ^  ^.  ^  .V.  o-  --  \ 

"^^^  ^']J'Zr^M~U^~  2  .  4  .  A^~"2.4.6.A6~^^^-^ 

But 

Ay 


:B 


VK^—x^ 
hence 

^^j<:^  <?*;<:*  y^x^ 

Integrating  each  t?erm  separately  we  have 

Taking  the  integral  between  the  limits 

x=-o  and  x=^A. 

we  have  for  half  the  surface  of  the  ellipsoid 

e"  c  "^  c 

S  =  27rBA(i- ^^— 7-etc.) 

^        2.3       2.45      2.4.6.7 


INTEGRAL    CALCULUS. 


335 


and  multiplying  this  expression  by  2  we  have  the  measure 

of  the  entire  surface  of  the  ellipsoid. 

If  we  make  A=B,  then  e=^o,  and  we  have  for  the  surface 

of  the  sphere 

S=4-R2 

as  before. 

(243)    To  find  the  surface  of  a  Paraboloid  of  revolution. 
We  have  (Art.  69)  in  the  case  of  the  paraboloid 

Integrating  according  to  (Art.  164)  we  have 

Estimating  from  the  vertex  where  S=o  andj^^o  we  have 


hence 


and 


3/  3 


C  =  — ^ 


^=Yp^{y'-\-p'Y-P^) 


(244)   To  find  the  area  of  the  surface  described  by  a  Cycloid 
revolving  about  its  base. 

We  have  (Art.  129)  in  the  case  of  the  cycloid 

ax=    , =^ 

V  2ry—y^ 

hence 


^,,'-+df  =/v/^.  +df-  =dy\/^. 


33^         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 


and  by  substitution 


^S  =  2 TcyV dx^  -\-dy^'=-2 ^ydy\ / ^^— 

V     27^ — J'^ 


or 

f^dy 


\-V  2r  f-^ 


V  zry—y^ 
But  we  have  found  (Art.  205) 


/ 


3 


y^dy  Sr     , 2y 


—  V  2)'  —  V — — V  2r~ 


y 


V2ry-y-  3       ^        •"       3 

hence 

/ 8^    / 2y    / 

S  =  2-V  2r  (»- — V  2r—  y— — V  2r—y)-j-C 

If  we  estimate  the  surface  from  the  plane  passing  through 
the  middle  point  of  the  base  we  shall  have  S=o  when  jj/  =  2r, 
hence  C=o.  Then  making  y=o  we  have  for  half  the  sur- 
face required 

(0,^     
— V  2r)^=^iTr^ 

and  for  the  entire  surface  twice  that  quantity.  That  is,  the 
area  of  the  surface  described  by  the  revolution  of  a  cycloid 
about  its  base  is  equal  to  twenty-one  and  one-third  times 
that  of  the  generating  circle. 

(245)  T/ie  Cubctture  of  Solids. 

The  cubaiure  of  a  solid  is  to  find  the  dimensions  of  an  equiv- 
alent cube  or  other  known  volume. 

We  have  (Art.  71) 

-y^dx 

for  the  differential  of  a  solid  of  revolution  where  y  is  the 

ordinate  and  x  the   abscissa  of   the   bounding  line  of  the 

revolving  surface  which  generates  the  solid;  and  the  axis  of 

abscissas  is  the  axis  of  revolution.     Hence 

N'=-f~y'^dx 


^— -r.v  and  _);2— -— ^3 


INTEGRAL   Cx\LCULUS.  337 

To  apply  the  formula  to  any  particular  solid  or  volume,  we 
eliminate  one  of  the  variables  by  means  of  the  equation  of 
the  bounding  curve,  thus  producing  a  differential  function 
of  one  variable  which  may  be  integrated. 

(246)  To  fi7id  tJie  volume  of  a  Right  Cone. 
Making  the  vertex  the  origin  we  have 

for  the  equation  of  the  bounding  line ;  but  a  is  the  tangent 

of  the  angle  made  by  .this  line  with  the  axis  of  the  cone,  and 

b      ^  , 

is  equal  to  y,  where  b  is  the  radius  of  the  base  and  h  the 

the  length  of  the  axis  ;  hence 

b  o      b'' 

^.vand;;'-^ 

whence 

,  b'^  />>-      x"^ 

N=-Jry-dx=-J--j^X"dx--—  .  ^;-+C 

Estimating  from  the  origin  we  have  V=o,  .t=o,  and  hence 
C=o,  and  making  x'-=^h  we  have  for  the  entire  cone 

h 

that  is,  the  volume  of  a  cone  is  equal  to  one-third  of  the 
product  of  its  base  by  its  altitude,  or  equal  to  one-third  of 
^  a  cylinder  of  the  same  base  and  altitude. 

(247)  To  find  the  volume  of  a  Sphere. 

From  the  equation  of  tlie  circle  we  have 

y-  —  R-  — .r- 
hence 

Estimating  from  the  plane  passed  through  the  center,  where 


;^^8        MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

x=Oy  we  have  S=o  and  0^=0,  and  making  ^=R  we  have 
for  half  the  volume  of  the  sphere 


V=|7rR3 


and  for  the  entire  sphere 

V=f;rR3 

Since   the  surface  of  the  sphere  is  equal  to  47rR.2  ^yg  have 

the  volume  equal  to  the  surface  multiplied  by  one-third  of 

the  radius. 

(248)  To  find  the  volume  of  an  Ellipsoid. 

Taking  the  origin  at  the  extremity  of  the  transverse  axis 
we  have  for  the  equation  of  the  bounding  line  or  curve 


/"B^  B^  x^  \ 

Y=/7zy^dx=7:J  j^{2Ax-x^)dx=7z-j{Ax'^  —  )-{-C 


A' 
hence 

B2  B2 

j^2\^Ax     X  )dx      >c^2^^i^ 

Estimating  from  the  origin  where  x^^o  we  have  V=o,  and 
hence  C=o ;  and  making  ^=2  A  we  have 

V=^|^(4A3-|A3)=4B2A=f:rB2  .  2A 

that  is,  the  volume  of  a  prolate  ellipsoid  is  equal  to  two- 
thirds  of  the  volume  of  a  cylinder  having  the  minor  axis  for 
its  diameter,  and  whose  altitude  is  equal  to  the  major  axis. 

If  the  ellipse  is  made  to  revolve  about  its  minor  axis  we 
should  have 

V'=7:|A2B 

for  the  volume  of  an  oblate  ellipsoid,  and  hence 

V:  V'::r|B2A:-fA2B::B:  A 
that  is,  the  volume  of  a  prolate  ellipsoid   is  to  that  of  an 
oblate  ellipsoid  generated  by  the  same  ellipse,  as  the  minor 
axis  is  to  the  major  axis. 


INTEGRAL    CALCULUS. 


339 


If  A=B  we  have 
as  before. 


V-tr:R3 


(249)  To  find  the  volmne  of  a  Paraboloid. 

In  "^his  case  we  have 


and 


N  ^1  -y'^  dx^=^2-pfxdx'=2r^p-—  ^~px^-{-C 


2 

Estimating  from  the  vertex  where  x=o  we  have  V=o,  and 
hence  C=^o,  and 

^  2 

or,  the  volume  of  a  paraboloid  is  equal  to  half  the  volume 
of  the  circumscribing  cylinder. 

(250)    To  find  the  volume  of  a  Solid  described  by  the  revolu- 
tion of  a  cycloid  about  its  base. 

Since  in  the  case  of  the  cycloid 

ydy 


dx- 


V  2ry — ■  j/^ 
we  have 


v-y-j''^^/A^=-/: 


y^ay 


V  2ry^y^ 
But  we  have  found  (Art.  204) 

/      /  -■— Xo— — y  2ry—y^ 

v  y  / 

Xo=^^X,--V2;t->'2 

2^2  -^ 


Xj=Xq      y  2ry — ^_y~ 
Xq  =  arc  of  a  circle  of  which  r  is  radius  and^v  the  versed 
sine. 


34°         MEASUREMENT    OF    GEOMETRICAL    MAGNITUDES. 

Integrating  between  the  limits  y=^o  and  y^=2r  we  shall 
have  half  the  volume  required;  but  y=-o  gives  V=o  and 
C=o  and_>'  =  2/'  gives 

■zr  7,~r''^ 

X. .^2 V"    ■ — . 


and 


I* ^\  Tf  TT^  ^^ 


hence  the  entire  volume  is 

But  the  volume  of  the  circumscribing  cylinder  is 

Hence  the  volume  of  the  solid  generated  by  the  revolution 
of  a  cycloid  about  its  base  is  five-eighths  of  that  of  the  cir- 
cumscribing cylinder. 


Appendix. 


ANSWERS 
TO    EXAMPLES    FOR    PRACTICE, 


ARTICLE  lo. 

Ex.  4. — A71S.     dx—bdy. 
Ex.  5. — Ans.     {a^-b^dx. 
Ex.  6. — Ans.     {c—d)dy. 
Ex.  7. — Ans.     adx-\-b(/y-\-cdz. 
Ex   S.—Afis.     b^du-^c^dz. 
Ex.  9. — Ans.     a"bdx-\-rdy. 
Ex.  TO. — Ans.     a^dy—b^dx. 
Ex.  Ti. — Ans.     b{ady—cdx). 
Ex.  12— Ans.     c'^{bdx-\'adz). 

ARTICLE  13. 

Ex.  4.—A?is.  xdy+ydx—udz—zdu. 

Ex   S-—Ans.  T,adx—2xdy—2ydx. 

Ex.  6. — Ans.  2(7dy+s^lu- 

Ex.  'j.—A?is.  4ab(yzdx+xzdy  +  xydz) 

Ex.  S.—Ans.  bcdu  —  zdy—ydz. 

Ex.  g.—A7is.  4a(xdy-^ydx)+cdu. 

Ex.  10. — Ans.     —2bdu^cdy. 

3 


ANSWERS    TO    EXAMPLES    FOR    PRACTICE. 


Ex.  3. — Ans. 
Ex.  4. — Ans. 


ARTICLE  14. 

adx      cdy 


x"       y 
{b—a)dx 


Ex.  5. — Ans,     ^dx-\- 


Ex.  6. — Ans.     — 2i^x- 


{^x—cY 

2  {c—y)dx-\-  2xdy 


{u—d  )dy  —  {y  —  c)du 


{u-dY 

Ex.  7. — Ans.     2adx-\-i^y—c)dx-\-xdy. 

Ex.  8. — Ans.     4{zi—c)(i>dy  —  dx)-\-4(dy—x)du. 

Ex.  9. — Ans. 

—  {^—y)dx-\-{a—x)dy     d-\-7i     a  —  x     vdu  —  {d-^u)dv 

X  ~\~~j         X 


{b-yY 


b—y 


Ex.  10.     A71S.      —{xy  —  z)d7J-\-{a  —  v)(xdy-j-ydx—dz). 
Ex.  II. — Ans.     — ^{xdy-{-ydx){u—c)— T,xydu. 
Ex.  12. — Ans. 


{dz—dy){a  —  x)  —  {z—y)dx— 


{y—z)dx-\-(c—x){dy—dz) 
{y-zY 


ARTICLE  19. 

Ex.  3. — Ans.  2axdx—^dx^dx-\-dx. 

Ex.  4. — Ans.  {c-\-d){2ydy — 2xdx\ 

Ex.  ^. — Ans.  2^x^dx — 2ady. 

Ex.  6. — Ans.  {nx'^~^  —  7,x^)dx. 

Ex.  7, — Ans.  ;^ax^dx  —  ;^bdx. 

Ex.  8, — Ans.  (x^  -{-a)dx-\-2x{x—a)dx. 

Ex.  9. — Ans.  2xy^dx-^2x^ydy — 22:^/2. 


ANSWERS    TO    EXAMPLES    FOR     PRACTICE. 


Ex.  lo. — A?is.     2ax{x'^ -\-a)dx-\-2iax'^dx. 


Ex.  II. — Ans. 
Ex.  12. — Ans. 
Ex.  i^.  —  Ans. 
Ex.  14. — Ans. 
Ex.  15. — Ans. 


4ayiiy 


xdx 


{a-\-x\ix 
\/  2ax—x2 
xdx 


{,-x-^y 


dx 


{x-\-\/i-x2yVl-x2 


Ex.  i6.-Ans.     3('^  +  Vx)V^ 

2\/  X 

Ex.  17. — Ans. 

2x{a'^ -\-a^ x'-^  +  x'^)d.\  ~\-{a'^  —  x~){ 2a^ x -\- j^x^)dx 
{a"^  -{-a^x^-\-x'^)''^ 

nx^^~^dx 


^xdx 
{i-x-^)^ 

2dx 


Ex.  18. — Ans. 
Ex.  19. — Alts. 

Ex.  20. — A?is.        ,         — ,    , ,     , xo 

V  I +  — ^2(^y  I +x  +  y  i—^j-  _ 

Ex.  2 1 . — Ans.     ;;;;/(^7+/^ji;" )"^~^  bnx'^~'^dx, 

■7(1  x"  dx 
Ex.  22. — Ans. ■=^ 

2\/ ax^  . 

-1   -4-  1    -_3 

Ex.  23. — A71S.     y^x  ^y  '^dx—\x^y  '^dy. 

1 
Ex.  24. — Ans.      —mx~^dx-\-p2(x^y)^(2xydx-\-x^dy). 

Ex.  25. — ^;/j-.      2xy^dx-i-lx''^y   ^dy—dz. 

^a\  26. — Ans.     bcdx  —  \y  '^dy. 


6  ANSWERS    TO    EXAMPLES    FOR    PRACTICE. 

Ex.  27. — Ans.      — '^x~^dx—y~^dy-\-2,z'^dz. 

Ex.  28. — Ans.     j(ax—y)  ^{adx—dy). 

_i 
Ex.  29. — Ans.     i(^— 0(-^~j)   ^{dx—dy). 

3 

i^:r.  ^o. — Ans.      .—    . 

ARTICLE  24. 

I          2            "^      „       4      „ 
^Jt:.  c. — Ans.     -^——^x+^x^ r^   +,  etc. 

^Jt:.  6. — Ans.     a^-\-l-a  ^x—^a   ^^^^ +Tf 2^   ^x'-^  —  ,etc. 
Ex.  y.—Ans.     a+il^a-'^x-iPa^x- +^%Pa-^x^-,  etc. 

Ex.  S.—Ans.     -  +i^--3j;+|r-V'  +i|^~ V^+,  etc. 

^jt:.  9.— ^/zi-.  ^"i+|^""'^":r2+|^"''3-^4  4.4o^- 3-^6_|_^  etc. 

ARTICLE  26. 
^:r.  4 .— ^;zj.      ^^  -  ;2.r^V  _|_;^(/^- 1)^,^2^ 2 

;/(;/— 1)(;2  — 2)        o   ^  ,      ^ 
^  x^    7  +  etc. 


2  •  3 

^^.5. — Ans.     x^—^x^y+lx  ^y^ -^f-^x  ^y^—Qtc. 
Ex  6. — Ans.     x'^-\-x'-y-\-x"^y^  +x''^y^-\-Q\.c. 

Ex.  T.— Ans.  a(x  ^-}-^x"iy-i-{^x"^'y^ +j\\x'^'^\^ -hetc.) 


ARTICLE  32. 

Ex.  8. — Ans.     A  maximum   when   x  =  —  'j;    a   minimum 
when  x=^  —  ^ 


ADDITIONAL    EXAMPLES    FuR    PRACTICE. 


.     .  h 

Ex.  o — Ans.     A  minimum  when  x=— . 

^  2 

Ex.  lo. — Ans.     A  maximum  when  .a== — . 

2C 

rx.  1 1 . — Arts.     A  minimum  when  x  =  ±  — . 

3^ 

2b 

Ex.  12. — Ans.     A  maximum  when  x=^ — . 

3^ 


ARTICLE  33. 

Ex.  13. — Ans.  The  height  of  the  cyHnder  is  equal  to 
the  diameter  of  the  base. 

Ex.  14. — Ans.  The  base  of  the  triangle  will  be  four- 
thirds  of  the  abscissa  of  the  extreme  point. 


ADDITIONAL  EXAMPLES  FOR  PRACTICE. 

DIFFERENTIALS    OF    FUNCTIONS. 

^x.  I.     What  is  the  differential  of  the  function 
z^  =^x{a -\-x){a^ -\-x'-^ )  ? 

Ans.     du=^((i^  -\-2a"x-\-2,ax^  \-4X^){/x. 

ix.  2.     What  is  the  differential  of  the  functio 

7^  =  {a-\-bxy^{7n-\-nxY  ? 
Ins.     du=^yi{a-\-bxY  {/n  +  //x)'^(/x-\-2bia-i-bx)  {m-\-nx)^^x 

Ix.  3.     Differentiate  n  —  {a-\-bx^Y{c-{-ex^Y 
lA?is. 
\iN  =  2o{a  +  bx-f{c  +  cx'^Yex^c/x  +  6b{c+cx^Y{a-\-bx^Yxdx 


S  ADDITIONAL    EXAMPLES    FOR    PRACTICE 

£x.  4.     Differentiate  2i=(a-\-\/lcY 

2V  X 


Ans.     ^^=na^V^)Vx 


Ex.  5.     Differentiate  u-=^x\a^-\-x'^)\/ a'^—x'^ 


Ans.     du 
Ex.  6.     Differentiate  u 


^a2—x2 

(I-\-X 


V  a — X 

.              ,          ■X'^—x 
Ans.     du  = dx 

2{a—x)^ 


Ex.  7.     Differentiate  u^=^~ 


X 


A/IS.     du 
Ex.  8.     Differentiate  u'=a 


X-]-^  I— x2 

dx 


\^  l—x'^    -\-2x{l—X") 

6(1 — x'^)dx 


Ans.     du- 


(34-^-)2y^ 


Ex.  9.     Differentiate  u=^ 


]ix'^~'^dx 


Ans.     dzi'=- 


Ex.  10.     Differentiate  ?<!=(^—:r)V'a2  4-^:3 

(a^—ax-{^2X^)dx 
Ans.     du=^ .  

Ex.  II.     Differentiate  u^={a^—x^)^  a-\-x 

Ans.     dz^=—  "-^ 


2\/  a  +  x 


Ex.  12.     Differentiate  u  — 


'\/x^  +JJ/3 


,             axydx — ax^dy 
Ans.     du=^—- ^ — - 


-    ADDITIONAL    EXAMPLES    FOR    PRACTICE. 


3 

Ex.  13.     Differentiate   ?^  =  (2a^-|-3^^)(a^— ^^)^ 


Ans.     du=^  —  i^x'^'\/a-2—x-2.dx 


Ex.  14.      Differentiate  u^=-  


'^ a  ■\-x — ^ a — X 

adx 
Ans.      (hi  =  — 


^  ^.„  .  a-}-2/^x 

Ex.  iK.     Differentiate  7^=-, — ,    ,    ,^ 


a\/a2 — x2—a^  -{-x'' 


.  ,       —2b^xdx 

Ans.     du  = 

{a+bx)^ 


DEVELOPMENT    INTO    SERIES. 

Ex.  16.     Develop  into  a  series  the  function 


ns.     i^=^a-i — -— :.  4- ,  —  etc. 

2a      2  .  4(1"^      2.4.  ba^ 

Ex.  17.      Develop  into  a  series  the  function 


U — \/  2X — I 

Ans.     7^=^y'  —i(i-\-x -f- —  —  etc.) 

22 


Ex.  18.      Develop  into  a  series  the  function 

I 


y  l?2—x2 

A71S.     7i^b-^-^U-^x^--^^~^b-^x^  +  ^  •  ^     ^^-''.yg+etc. 

2.4  2.4.6 

Ex.  19.     Develop  into  a  series  the  function 

LO  4  K      2    —2-  c;       2       I    — 8- 

A71S.     u=a  ^  4-4t?^A-4-^-T^  ^a-"^— ^-^ — '—a  ^a'^+etc. 

3.6  3-6.9 


lO  ADDITIONAL    EXAMPLES    FOR    PRACTICE. 

Ex.  20.     Develop  into  a  series  the  function 

I 
u—- 


Ans. 
a      4a^     4  .  8^^     4.8.  12^^  ^     4.8.12.  16^^ '' 


MAXIMA    AND   MINIMA. 

^.^.  21.     Find  the   values  of  x  which   will   render  u  a 
maximum  or  a  minimum  in  the  equation 

u=^x^  —  Sx^  +  2  2Ji;^  —  24X-\-l2 

Alls.     A  maximum  when  x  =  2\    a  minimum  when  x  =  i 
or  3. 

Ex.  22.     Find   the   values  of  x  which   will  render  ti  a 
maximum  or  minimum  in  the  equation 

?/  =  3.T^  —  16.1;^  •^Gx'^  -\-']2X —  I 

Ans.     There   is   a   maximum  when  x=^2  and  a  minimum 
when  x=^  —  i  or  -{-3. 

-fi"^.  23.     Find  the  maximum  or  minimum  values  of 

a'^x 
\a—x)~' 
Ans.     There  is  a  minimum  value  of  u  when  x^=^—a. 
Ex.  24.     Find  the  maximum  or  minimum  value  of 

?^=4.T^  —  T,X 
Ans.     There  is  a  maximum  when  a^=|-. 

Ex.  25.     Find  the  fraction  that  exceeds  its  cube   by   the 
maximum  quantity. 

Ans.      H -^ 

V  3 

Ex.  26.     What  is  the  greatest  positive  value  of  x—x"^  ? 

1 
J' 


ADDITIONAL    EXAMPLES    FOR    PRACTICE.  II 

t.  27.  The  hypothenuse  and  one  side  of  a  triangle  are 
together  equal  to  18  inches.  What  is  the  length  of  each 
when  the  triangle  is  a  maximum  ? 

A?is.     Hyp.  12  in.;  side,  6  in. 

Ex.  28.  Required  the  minimum  triangle  formed  by  the 
radii  produced  and  the  tangent  to  the  quadrant  of  a  circle? 

Afis.     The  triangle  is  isosceles. 

Ex.  29.  What  value  of  x  will  render  ^']V x—x  a  maxi- 
mum? AflS.       Sl2\. 

Ex.  30.  A  vessel  is  on  the  equator,  100  miles  west  of  a 
certain  meridian,  sailing  east  at  the  rate  of  7  miles  per 
hour;  another  vessel  is  on  that  meridian,  50  miles  north  of 
the  equator,  at  the  same  moment  sailing  south  at  the  rate 
of  10  miles  per  hour  ;  where  will  the  vessels  be  when  near- 
est to  each  other  ? 

Ans.  The  first  will  be  43.63  miles,  nearly,  west  of  the 
meridian,  and  the  other  30.5  miles,  nearly,  south  of  the 
equator;  and  the  distance  between  them  will  be  53.14 
miles,  nearly. 

Ex.  31.  Three  towns  are  at  the  angles  of  a  right-angled 
triangle;  A  and  B  at  the  base,  which  is  25  miles  long  from 
east  to  west,  and  C  100  miles  north  of  A.  There  is  a  rail- 
way from  A  to  C,  on  which  the  cars  go  through  in  5  hours. 
A  man  at  j5  wishes  to  reach  C  in  the  shortest  time,  and  can 
travel  5  miles  per  hour  on  horseback.  He  decides  to  ride 
to  the  railway  and  take  the  cars.  At  what  point  must  he 
meet  the  train  in  order  to  accomplish  his  ol^ect  ? 

Ans.     At  a  point  — ^  miles  from  A. 

V  15 

Ex.  32.  Let  there  be  two  lines,  each  equal  to  ^,  stand- 
ing at  the  extremities  of  a  third  line  a,  and  equally  inclined 
to  it;  what  must  be  the  length  of  the  fourth  line  C,  connect- 


12  ADDITIONAL    EXAMPLES    FOR    PRACTICE. 

ing  the  upper  extremities  of  the  lines  b^  so  that  the   area  of 
the  trapezium  thus  formed  shall  be  a  maximum  ? 

Ans.     C=-+\/2^^+— 

2  4 

Ex.  i^i-  ^^'^^  two  sides,  each  equal  to  b,  of  an  isosceles 
triangle  being  given,  what  must  be  the  length  of  the  third 
side,  so  that  the  area  of  the  triangle  shall  be  a  maximum  ? 

Ans.     V  2b^ 

Ex.  34.  What  decimal  fraction  exceeds  its  cube  more 
than  any  other  numerical  quantity  can  exceed  its  cube  ? 

Ans.     .577  + 

Ex.  35.  Divide  25  into  two  such  parts  that  the  product 
of  the  second  power  of  one  part  by  the  third  power  of  the 
other  may  be  a  maximum ;  and  which  part  must  be  cubed  ? 

Ans.     The  numbers  are  15  and  to. 
The  number  15  must  be  cubed. 

Lx.  2i^.  What  will  be  the  proportions  of  the  largest  rec- 
tangle that  can  be  inscribed  in  an  ellipse  of  which  2A  is  the 
transverse  axis  and  2B  the  conjugate  "i 

A71S.     The  sides  of  the  rectangle  are  A\/  2    and  3^/2 

Ex.  37.  What  is  the  height  of  the  smallest  cone  that 
will  enclose  a  sphere,  lying  on  the  plane  of  its  base? 

Ans.     Four  times  the  radius  of  the  sphere. 

Ex.  38.  What  are  the  greatest  and  least  values  of  the 
ordinate  of  the  curve  represented  by 

a^y^^ax"^  — x^ 

Ans.  When  x^=-\a.,y  is  a  maximum;  and  when  x=-o., y 
is  a  minimum,  being  also  equal  to  zero. 


ADDIllONAL    EXAMPLES    FOR    PRACTICE. 


^3 


SURTANGENTS    AND    SUBNORMALS. 

Ex.  39.      What  is  tlie  subnormal  to  the  curve  whose  equa- 
tion is  y^~2<i;-  /(?<,'■.  .r 

Ans. 

X 

Ex.  40.     What   is   the   value    of    the   subnormal   of    the 
curve  wliose  equation  is 

2ay~  A;-a'^  =2a-^ 

3-v2 

Ans.     '- 

2a 

Ex.  41,     What    is    the    value   of  the   subtangent   of  the 

curve  whose  equation  is 


.3 


y       a —  X                    1           zx(a  —  x) 
"     "^  A/iS.      : 

3(7 — 2X 

Ex.  42.      Required   the   subtangent  of  the   curve   whose 
eiiuation  is  xv-=^a''^{(7  —  .v) 

Ans.      — ^ 

a 

Ex.  43.     When  is  the  subtangent  of  the  preceding  curve 

a  minimum  ?  Ans.     When  x^^hi 

Ex.  44.      Required   the   value   of  the   subtangent   of  the 
curve  whose  equation  is 

x'^y^  =^{a-\-x)~{/f~  — .r-) 

_l^^^  _x{c7-{-x){P—x') 

CURVATURE    AND    CURVED    LINES. 

Ex.  45.     Find  the  radius  of  curvature  at  any  point  of  the 
cubical  parabola  whose  equation  is 

A;is.  yc= T^. 

oa'v 


14  ADDITIONAL    EXAMPLES    FOR    PRACTICE. 

Ex.  46.     When   is  the   curvature  of  the    preceding  curve 
greatest  ? 

4  /     Y' 
Ans.     When^v=/\/    ^^ 

40-5 

^.T.  47.     Find   the   radius  of  curvature    at   any   point  of 

the  logarithmic  curve  whose  equation  is 

Ajis.     R  — -ir^ — - — ,  J/bemg  the  modulus  and  ^r  the  base. 

Ex.  48.      Find    the    point   of    greatest   curvature    of   the 
logarithmic  curve. 

^,  •  •  .  M 

Ans.     The  point  whose  ordinate  is  equal  to 


V 


Ex.  49.     rind  whether  the  curve  whose  equation  is 


y^  ^=-x^ 


has  a  point  of  inflexion. 

Ans.     The  curve  has  a  point  of  inflexion  at  the  origin. 

Ex.  50.      Find  the  point  of  inflexion   in   the  curve  whose 
equation  is  ax^-=^a^y^x'-^y 

Ans.     There  is  an  inflexion  at  the  point  where  y^^-^a. 

Ex.  51.     Find  the  point  of  inflexion   in   the  curve  whose 

equation  is  x'^y^^^^a^iax — x^^ 

Ans.     Thcie  are  two  points  of  inflexion  corresponding  to 

3^  (? 

x^=-- — -,  and  y=  +     ,^ 
4  -"      -V3 

Ex.  52       Find  whether  the  curve  whose  equation  is 

lias  a  cusp  at  the  point  where  the  tangent  is   parallel  to  the 
axis  of  ordinates. 

Ans.     There  is  a  cusp  of  the  first  order  at  the  point  where 
y^=b  and  x^=a 


INTEGRAL    CALCULUS.  I5 


INTEGRAL  CALCULUS. 

INTECRATION    OK    DIFFERENTIALS. 


Ex.  I.      Integrate  the  differential 

dx 
du 


{a  —  xY^ 

Ans.     21 


^[a  —  xY 

Ex.  2.      Integrate  the  differential 

^xdx 


dlt  = 


(l  — JC-)* 


2 

Am.     u 


I — x"^ 


Ex.  3.      Integrate  the  differential 

ladx 
du- 


-^  V  2ax—  x^ 


^V  2iIX—X'^ 

Ans.     //  =  —  — 


X 


Ex.  4.     Integrate  the  differential 

x^'dx 

du ' 


Va-^-Jr  6x^ 
Ans.     u^=- 


-7 


Ex.  5.      Integrate  the  differentia. 

dx 

du^=-- 


V  i-{-x- 

X'^  ^7,X'^  3  .  5.T" 

Ans.      u=x-_-z  +  ^\     --->      .     r.     -+^^^' 


l6  INTEGRAL    CALCULUS. 

Ex.  6,     Integrate  the  differential 

2iX^dx 


du  —  -ir~. — I 
X'^  -f-  a"^ 

Ans.     u^log.  {x^-{-a'^) 


Ex.  7.      Integrate  the  differential 

^x'^dx 

3.v^  +  7 

Ans.     It  =j\  log.  (3:^^  +  7 ) 

Ex.  8.     Integrate  the  differential 

T,x'^  -\-2x-\-1 
du  =     7~;      5—^       —  (7''.v 
.:r'^  -\-X^  -tX-\-  I 

y///^\     ?/  ^log.  {x^-\-x'^  -\-x --f  - 1 ) 
^jt.  9.     Integrate  the  differential 

du  =  ■—, — , 

V  d-'-Vx- 
Ans.      u^=  — ;; 


DIRECT   METHOD   OF   KATES.  17 


«IOiIE  "WXOfPLES 

SHOWING  THE  APPLICATION 

OF   THE 

PRINCIPLE  SET  FORTH  IN  THIS  WORK. 

The  grand  instrument  by  which  the  ancient  mathemati- 
cians brought  the  science  of  Geometry  to  so  great  a  degree 
of  perfection  was  the  ^'method  of  e.rJiaustion^^^  or  otherwise, 
the  ''  reductio  ad  absurd  amy  When  the  science  awoke 
from  its  long  sleep,  by  the  brilliant  discoveries  of  Cavalieri, 
Pascal.  Descartes,  Leibnitz  and  Newton,  the  mathematical 
world  became  impatient  of  the  scrupulously  exact  but  tedious 
and  laborious  method  of  the  ancients,  and  demanded  shorter 
and  easier  methods.  Various  were  the  methods  resorted  to. 
The  indivisibles  of  Cavalieri  was  one  of  them,  being  pro- 
ncninced  by  Carnot  to  be  ''an  abbreviation  by  means  of 
which  we  obtain  promptly  and  easily,  in  many  cases,  what 
could  be  discovered  only  by  long  and  painful  processes 
according  to  the  method  of  exhaustion."  Such,  also,  Avere 
the  infinitesimals  of  Pascal  and  Leibnitz,  by  which  they 
converted  curves  into  polygons,  and  thus  avoided  the  "  re- 
ductio ad  absurdum,"  by  the  adoption  of  a  notion,  of  which 
the  absurdity  is  intrinsic  and  without  any  alternative.  Such 
also  was  the  object  of  the  celebrated  lemmas  of  Newton,  in 
the  first  book  of  his  Principia;  of  which  he  says:  ''These 
lemmas  are  premised  to  avoid  the  tediousness  of  deducing 


18  DIRECT   METHOD    OF   RATES. 

perplexed  demonstrations  ad  ahsurdiim  according  to  the 
method  of  the  ancient  geometers."  The  first  of  these  lem- 
mas (see  page  40)  seems  to  be  a  sort  of  generalization  of 
the  reductio  ad  absurdum  so  that  the  principle  may  be 
applied  at  pleasure  without  going  through  the  tedious 
formula  in  each  particular  case. 

Many  modern  geometers  have  adopted  the  infinitesimal 
method,  and  demonstrate  the  properties  of  the  circle,  for 
instance,  by  considering  it  a  polygon  with  an  indefinite 
number  of  sides.  Says  one  "^  "  the  circumference  is  the  limit 
of  the  perimeter  of  the  polygon,"  and  adds,  ''no  sensible 
error  can  arise  in  supposing  that  what  is  true  of  such  a 
polygon  is  also  true  of  its  limit,  the  circle."  But  can  an]/ 
error  arise  ?  If  so,  then  the  polygon  does  not,  strictly  speak- 
ing, coincide  with  the  circle. 

Another  author  has  struck  out  a  different  method.  He 
says,  indeed  truly,  that,  f  "  strictly  speaking,  the  circle  is  not 
a  polygon,  and  the  circumference  is  not  a  broken  line,"  and 
therefore  instead  of  the  method  of  infinitesimals  he  adopts 
what  he  terms  the  method  of  limits.  J  "The  principle," 
says  he,  "  upon  which  all  reasoning  by  the  method  of  limits 
is  governed,  is  that,  whatever  is  true  up  to  the  limit  is  true  fli* 
the  limit.  We  admit  this  as  an  axiom  of  reasoning  because 
we  cannot  conceive  it  to  be  otherwise."  Again,  ||  "  Now,  it 
is  evident  that  by  the  process  described  "  (bi-secting  the  sides 
indefinitely)  "the  polygons  can  be  made  to  approach  as  nearly 
as  we  please  to  equality  with  the  circle,  but  can  never  entirely 
reach  it."  Bearing  in  mind  this  last  admission,  how  shall 
we  construe  the  above  "axiom"?  How  can  we  follow  the 
polygon  up  to  the  limit  without  arriving  at  the  limit?  But 
if  "  up  to  the  limit  "  means  anything  short  of  "  at  the  limit," 
then  truly  the  polygon  does  not  become  the  circle,  and  '.ve 

*  Davits'  Legendrc,  revised  edition  of  1856. 

t  ltay'8  I'liine  and  Solid  Geometry,  Art.  477.  +  Art.  201.  ||  Art.  47-'=. 


DIRECT    METHOD   OF   KATES.  19 

must  deny  that  "whatever  is  true'''  (of  the  polygon)  "  up  to 
the  limit,  is  true  "  (of  the  circle)  ''  at  the  limit/'  We  say 
"of  the  circle  "  because  the  circle  is  the  limit  of  the  polygon. 
In  this  case  all  the  original  differences  between  the  polygon 
and  the  circle  remain  intact.  The  boundary  of  a  pol3^gon  i:^ 
a  broken  line,  that  of  a  circle  is  not.  The  periphery  of  a 
polygon  has  angles,  that  of  a  circle  has  none.  The  area  of 
the  polygon  is  less  than  that  of  the  circumscribed  circle, 
which  is  not  true  of  the  circle  itself. 

"The  author  does  not  claim  tl"^  credit  of  having  discovered 
or  invented  this  new  axiom.  ''In  explaining  the  doctrine  of 
limits,"  says  he,  '  the  axiom  by  Dr.  Whewell  is  given  in  the 
words  of  that  eminent  scholar.'  Now,  Dr.  Whewell  certainly 
had  no  use  whatever  for  any  such  axiom.  For,  according  to 
his  view,  the  variable  magnitude  not  only  approaches  as  near 
as  we  please,  but  actually  reaches  its  limit.  Thus,  says  he, 
^a  line  or  figure  ulthnately  coincides  with  the  line  or  figure 
which  is  its  limit.'  Now,  most  assuredly,  if  the  inscribed 
polygon  ultimately  coincides  with  the  circle,  then  no  new 
axiom  is  necessary  to  convince  us  that  what  is  always  true 
of  the  polygon  is  also  true  of  the  circle.  For  this  is  only  to 
say  that  what  is  true  of  the  variable  polygon  in  all  its  forms 
is  true  of  it  in  its  last  form.  According  to  his  view,  indeed, 
there  was  no  chasm  to  be  bridged  over  or  spanned,  and 
consequently  there  was  no  need  of  any  very  great  labor  to 
bridge  it  over  or  to  span  it.  The  truth  is,  however,  that 
although  he  said  the  two  figures  would  ultimately  "coincide," 
leaving  no  chasm  between  them  to  be  crossed,  he  felt  that 
there  would  be  one,  and  hence  the  new  axiom  for  the  purpose 
of  bridging  it  over. 

But  if  we  reject  these  methods  of  getting  rid  of  the 
"reductio  ad  absurdum,"  what  shall  we  do?  "If  we  deny 
that  the  pol3^gon  and  the  circle  ever  coincide,  how  sliall  we 


•  Bledsoe's  Philosophy  of  Jlatliiiimtics. 


20  DIRECT   METHOD   OF   RATES. 

l)ridge  over  the  chasm  between  them  so  as  to  pass  from  the 
knowledge  of  right  lined  figures  and  volumes  to  that  of 
€urves  and  curved  surfaces?  Shall  we,  in  order  to  bridge 
over  this  chasm,  fall  back  on  the  reductio  ad  absurduin  of 
the  ancients?  or  can  we  find  a  more  short  and  easy  passage 
without  the  sacrifice  of  a  perfectly  logical  rigor  in  the  transit? 
This  is  the  question.  This  is  the  very  first  problem  which  is 
and  always  has  been  presented  to  the  cultivators  of  the 
modern  methods.  Is  there,  after  the  lapse  and  labor  of  so 
many  ages,  no  satisfactory  solution  of  this  primary  problem? 
It  is  certain  that  none  has  yet  been  found  which  has  become 
general  among  mathematicians."  I  believe,  moreover,  that 
«uch  a  method  never  will  be  found.  But  I  also  believe  that 
no  ''bridge"  is  necessary  —  that  instead  of  starting  with  the 
polygon  and,  with  it,  trying  to  reach  the  circle,  we  may  go 
directly  to  the  circle  itself  and  boldly  compel  it  to  yield  up 
its  properties  to  the  open  light  of  mathematical  truth  without 
any  "sacrifice  of  perfectly  logical  rigor." 


DIRECT    METHOD   OF    RATES.  21 


THE 

DIRECT  METHOD  OF  RATES 

AS    APPLIED    TO 

PLANE  AND  SOLID  GEOMETRY. 

Several  of  the  propositions  in  Elementary  Geometr}'  have 
never  been  solved  directly.  The  result  has  been  obtained, 
either  by  the  '^  reductio  ad  ahsurdum,'^  which  is  logical  and 
conclusive,  although  indirect;  or  else  by  the  method  of  limits, 
or  infinitesimals. 

I  propose  to  apply  the  direct  method  of  Bates  to  some  of 
these  propositions,  by  way  of  example  and  illustration.  For 
this  purpose  we  assume  the  following  fundamental  princi- 
ples or  axioms. 

Axiom  I. 

The  rate  of  incf^ease  of  ant/ geometrical  magnitude  ivhile  be- 
ing generated,  may  he  measured  at  any  moment,  hy  a  supposi- 
tive  uniform  increment  that  would  arise  if  the  generatrix  were 
to  continue  to  move  with  the  existing  velocity,  measured  in  a 
direction  perpendicular  to  itself^  for  one  unit  of  time  without 
changing  its  own  magnitude. 

Thus  a  right  cone  may  be  generated  by  its  base  flowing  up- 
ward and  gradually  contracting  so  as  to  bring  it  to  a  point  at 
the  vertex  of  the  cone.  But  the  rate  at  which  it  might  be 
increasing  at  an}-  instant  would  be  represented  b}'  the  cylin- 
der that  would  be  formed  if  the  generatrix  were  to  continue 


22  DIRECT   METHOD    OF   RATES. 

its  movement,  without  contraction,  for  one  unit  of  time  ;  for 
the  base  of  this  cylinder  would  represent  the  generatrix  and 
its  height  would  represent  its  velocity — and  these  two,  to- 
gether, compose  the  rate. 

Axiom  II. 

If  two  magnitudes  begin  to  he  at  the  same  moment,  and  the 
ratio  of  their  rates  of  increase  is  constant,  the  ratio  of  the 
magnitudes,  themselves^  ivill  he  constantly  the  same  as  that  of 
their  rates. 

Thus,  if  two  persons  set  out  at  the  same  moment  and 
place  to  travel  in  the  same  direction,  at  constant  rates,  the 
ratio  of  the  distances  traveled  by  each  will  be  constantly  the 
same  as  that  of  their  rates  of  travel. 

Axiom  III. 

//  a  geoinetrical  magnitude  he  generated  at  a  uniform  rate^ 
the  quantity  generated  tvill  he  proportional  to  the  time  occu- 
pied. 

This  axiom  needs  no  illustration. 

Proposition"  I. 

Two  angles  are  to  each  other  as  the  intercepted  arcs,  de- 
scribed from  their  vertices  with  equal  radii. 

Let  A  C  B  and  D  C  B  be  the  two  given 
angles,  and  A  B  and  D  B  the  correspond- 
ing arcs.  Since  the  arcs  are  described  with 
equal  radii  they  may  be  considered  as  parts 
of  the  same  circumference,  and  the  angles 
as  being  at  the  center  of  the  same  circle,  as 
in  the  figure.  Let  us  suppose  the  radius 
C  A  to  revolve  around  the  center  C  at  a  uniform  rate,  thus 
describing  the  angles  and  the  arcs.  Then  it  is  evident  (Ax- 
iom III)  that  the  arcs  will  be   to  each   other  as  the  times  in 


DIRECT    METHOD    OF   RATES.  23 

which  they  are  described,  and  the  angles,  being  described 
during  the  same  times  will  be  in  the  same  ratio.  Hence,  the 
angles  Avill  be  to  each  other  as  the  corresponding  arcs. 

PROPOSITIOlSr   II. 

Rectangles   having   the  same  altitude  are  to  each  other  as 

their  bases. 

Let  A  E  and  B  F  be  the  two  rectan- 
gles, whose  bases  may  be  placed  on  the 
same  line,  as  in  the  figure.  Then  B  E 
will  be  the  height  common  to  both. 
Suppose  the  rectangle  A  E  to  be  gen- 
erated by  the  line  B  E  moving  towards 
A  D  at  a  uniform  rate  and  constantly  parallel  to  itself ;  and 
the  rectangle  B  F  to  be  generated  by  the  same  line  B  E  mov- 
ing in  like  manner  toward  C  F  at  precisely  the  same  uni- 
form rate.  Then  (Axiom  III)  the  areas  A  E  and  B  F  Avill  be 
to  each  other  as  the  times  required  to  described  them  ;  and 
the  bases  of  the  rectangles,  which  measure  the  distances 
passed  over  by  the  generatrix  in  each,  being  described  at  a 
uniform  rate,  will  be  proportional  to  the  same  times.  Hence, 
the  areas  of  the  two  rectangles  are  to  each  other  as  their 
bases. 

Proposition  III. 

Two  rectangular  ixwaUeloplpeds  having  equal  bases  are  to 
each  other  as  their  altitudes. 

Suppose  the  volumes  to  be  described  or  generated  by  their 
bases  flowing  upward  at  equal,  uniform  rates  ;  then  (Axiom 
III)  the  volume  generated  in  each  case  will  be  proportional 
to  the  time  occupied  b}"  the  movement  of  its  generatrix  ; 
and  the  altitude,  which  measures  the  movement  in  each  case 
will  also  be  proportional  to  the  time.  Hence  the  volumes 
will  be  to  each  other  as  their  altitudes. 


24 


DIKECT   METHOD    OF   RATES. 


Proposition  IV. 

The  volumes  of  two  Pyramids  having  equivalent  bases  and 
equal  altitudes  are  equivalent  to  each  other. 

LetSABCandODEF 
be  two  pyramids  of  which 
the  bases  A  B  C  and  D  E  F 
are  equivalent  triangles  and 
the  altitudes  are  equal. 
Then  let  us  suppose  each 
of  these  pyramids  to  be 
generated  by  the  flowing 
of  its  base  upward  at  the 

same  rate  as  the  other,  in  ^  E 

a  vertical  direction,  and  contracting  its  sides  so  that  its 
angles  shall  always  be  in  the  edges  of  the  pyramid.  Then, 
since  the  rate  of  generation  of  the  pyramid  at  any  moment 
will  be  in  proportion  to  the  area  of  the  generatrix,  multiplied 
by  the  rate  at  which  it  is  moving;  and  since,  in  this  case,  the 
rates  are  the  same  and  the  generatrices  at  all  times  equiva- 
lent, the  rates  of  generation  of  the  two  pyramids  will,  at  all 
times,  be  equal;  hence  (Axiom  II),  the  magnitudes  of  the 
parts  generated  will  be  constantly  equivalent,  and,  of  course, 
the  pyramids,  when  completed,  will  also  be  equivalent. 

Proposition  V. 

To  find  the  area  of  a  Circle. 

We  will  suppose  the  circle  to  be  generated  by  the 
tion  of  radius  C  A  about  the  center  C  at  a 
uniform  rate.  When  the  radius  is  in  the 
position  C  A,  and  revolving  toward  B,  every 
point  in  it  will  tend  to  move  in  a  direction 
perpendicular  to  C  A,  and  hence  the  point  A 
will  tend  to  describe  the  line  AB  tangent  to  fio.  4. 

the   circle   and  perpendicular  to  C  A  ;  and  if  left  to  its  ten- 


DIRECT   METHOD   OF   RATES. 


25 


dency  ivould  describe  that  line  at  a  uniform  rate.  The  line, 
therefore,  may  be  taken  as  the  symbol  representing  the  rate 
of  increase  or  generation  of  the  circumference  of  the  circle. 
But  while  the  point  A  tends  to  move  in  the  direction  A  B, 
every  point  in  the  radius  C  A  tends  to  move  in  a  direction 
parallel  to  it,  and  at  a  rate  proportional  to  its  distance  from 
the  center  C.  Hence  the  radius  itself,  if  left  to  its  tendevcy 
when  at  C  A.  would  be  found  at  C  B,  when  the  point  A  is  at 
B  ;  and  the  triangle  CAB  would  be  generated  at  a  uniform 
rate  during  the  same  time  that  the  line  A  B  is  generated. 
The  triangle  may,  therefore,  be  taken  as  the  symbol  of  the 
rate  at  which  the  area  of  the  circle  is  generated,  and  the 
ratio  of  these  symbols  is  also  the  ratio  of  the  rates  which 
they  represent.  But  the  triangle  is  equal  to  i  C  A. .  A  B;  that 
is,  the  ratio  between  the  rate  of  generation  of  the  circumfer- 
ence and  that  of  the  area  of  the  circle  is  half  radius  ;  and 
this,  being  constant,  is  the  ratio  between  any  part  of  the  cir- 
cumference and  the  corresponding  part  of  the  circle  through- 
out their  generation,  and,  of  course,  of  the  entire  circum- 
ference to  the  entire  circle.  Hence  the  area  of  the  circle  is 
equal  to  half  the  radius  into  the  circumference. 

Proposition  YI. 
To  find  the  area  of  the  convex  surface  of  a  Cone. 
Suppose  the  cone  to  be  generated  by  the  revolution  of  the 
triangle  ADC  (Fig.  5)  about  the  axis  D  C.     The  hypothe- 
nuse  A  D  will  generate  the  convex  sur- 
face, and  the  point  A  will  generate  the 
circumference  of  the  base.     When  the 
triangle  is  in  the  position   ADC  and 
revolving  towards  E,  the  point  A  if  left 
to  its  tendency  at  that   instant  would 
describe  the  line  A  E,  perpendicular  to 
A  C,  in  some  unit  of  time,  and  hence  fig.  5. 

A  E  may  be  taken  to  represent  the  rate  at  which  the  circum- 


26  DIRECT   METHOD    OF   RATES. 

ference  of  the  base  is  generated.  Now  every  point  in  the 
line  A  D  tends  to  move  in  a  direction  parallel  to  A  E,  and  at 
a  rate  proportional  to  its  distance  from  the  axis  D  C  ;  hence 
if  left  to  that  tendency  the  line  A  D  would  describe  the  tri- 
angle A  D  E,  and  be  found  at  D  E  in  the  same  unit  of  time. 
Hence  A  D  E  (Axiom  I)  may  be  taken  to  represent  the  cor- 
responding rate  of  generation  of  the  convex  surface  of  the 

cone.   But  A  D  E  -:  A  E  .  —  or  ^i^A  .=  AP.  that  is,  the 

2  AE  2    ' 

ratio  between  the  rates  of  generation  of  the  convex  surface 
of  the  cone  and  the  circumference  of  its  base  is  constant  and 
equal  to  half  its  slant  height.  Hence  the  ratio  between  the 
magnitudes  generated  will  be  the  same  (Axiom  II),  and  the 
convex  surface  divided  by  the  circumference  of  the  base 
equals  half  the  slant  height,  or,  the  convex  surface  equals 
the  circumference  of  the  base  multiplied  by  half  the  slant 
height. 

Proposition-  VII. 

To  find  the  measure  of  tJie  volume  of  a  Cone. 

The  cone  being  supposed  to  be  generated  by  the  revolution 
of  a  right  angled  triangle  about  one  of  its  sides,  which  be- 
comes the  axis  of  the  cone,  while  the  base  is  generated  by 
the  other  side  as  its  radius,  let  us  suppose  the  generating 
triangle  to  have  arrived  at  the  position  ADC  (Fig.  5) — the 
point  A  moving  towards  E.  Then  every  point  in  the  trian- 
s\q  tends  to  move  in  a  direction  perpendicular  to  its  plane 
and  at  a  rate  proportional  to  its  distance  Irom  the  axis  C  D  ; 
so  that  if  A  E  is  taken  to  represent  the  line  that  would  be 
described  by  the  point  A  in  a  unit  of  time  in  consequence  of 
that  tendencij^  then  at  the  end  of  the  same  unit  of  time  the 
line  A  D  would  be  found  at  E  D,  and  the  triangle  A  D  C,  at 
E  D  C,  so  that  the  pyramid  D  A  E  C  would  be  the  volume  gen- 
erated b}^  the  triangle,  during  the  same  unit  of  time  and  may 


DIRECT   METHOD   OF   RATES. 


27 


therefore  (Axiom  1)  be  taken  to  represent  the  rate  at  which 
the  cone  is  generated  ;  while,  at  the  same  time,  the  triangle 
ACE  would  be  described  by  the  radius  A  C  of  the  base,  and 
would,  therefore,  represent  the  rate  at  which  the  base  of  the 
cone  was  generated.  But  the  volume  of  the  pyramid  D  AE  C 
is  equal  to  its  base  ACE  multiplied  by  one-third  of  its  alti- 
tude D  C.  Hence  the  rate  of  generation  of  the  cone  divided 
by  that  of  its  base  =  a  constant  quantity.  Therefore  (Axiom 
II)  the  cone  itself  divided  by  its  base  is  equal  to  the  same 
quantity,  being  one-third  of  its  height — or  the  volume  of  the 
€one  is  equal  to  its  base  multiplied  by  one-third  of  its  height. 


Lemma  I. 

To  find  the  area  of  the  convex  surface  of  an  ungula  ivith 
a  semicircular  base. 

An  ungula  of  this  kind  is  formed  by  passing  a  plane 
obliquely  through  the  center  of 
the  circular  base  of  a  right  cylin- 
der. The  ungula,  thus  cut  off, 
will  have,  for  its  lower  base,  half 
of  the  base  of  the  cylinder,  and, 
for  the  upper  one,  a  semi-ellipse, 
while  the  elements  of  its  convex 
surface  will  be  portions  of  the 
elements   of  the  cylinder.     The  L     ivi       c 

height  of  the  ungula  will  be  measured  by  that  element  of  its 
-convex  surface  which  is  ninety  degrees  from  the  extremities 
of  its  base. 

Let  E  D  A  B  be  the  convex  surface  of  the  ungula  of  which 
A  D  B  is  the  base  and  E  D.  perpendicular  to  the  base,  is  the 
height.  Then  suppose  the  convex  surface  to  be  generated  by 
an  element  F  I,  moving  parallel  to  itself,  with  its  foot  in  the 
circumference  of  the  circular  base,  and  its  length  varying  in 
such  a  manner  that   its  upper  extremity  shall  be  constantly 


28  DIRECT   METHOD   OF   RATES. 

in  the  elliptic  boundary  of  the  upper  base.  Draw  I  H  tan- 
gent to  the  base  at  the  foot  of  the  element  P  I,  F  G  parallel 
to  I  H,  G  H  parallel  to  F  I,  I  L  and  H  M  perpendicular  to 
the  diameter  A  B,  and  I  K  parallel  to  it.  Join  I  C  and  as- 
sume I  H  to  be  the  line  that  would  he  generated  by  the  foot 
of  the  element,  F  I,  in  one  unit  of  time,  if  left  to  its  ten- 
dency, with  the  velocity  it  had  the  moment  it  arrived  at 
its  present  position  ;  it  will  then  (Axiom  I)  represent  the 
rate  at  which  the  circumference  of  the  base  is  generated  at 
that  moment  ;  and  the  rectangle  F  G  H  I,  being  generated 
at  a  uniform  rate  in  the  same  unit  of  time,  will  be  the  sym- 
bol of  the  rate  at  which  the  convex  surface  of  the  ungula  is 
generated.  Also  L  M,  being  a  uniform  increment  of  the 
diameter  of  the  base,  generated  in  the  same  unit  of  time, 
will  represent  the  rate  at  which  that  diameter  is  generated. 
Since  every  plane  cutting  A  B  at  right  angles,  will  cut 
the  surfaces  of  the  ungula  in  similar  triangles,  there  will  be 
a  constant  ratio,  which  we  call  r,  between  each  element  of 
the  convex  surface  of  the  ungula  and  the  corresponding  or- 
dinate of  the  base  ;  hence  we  may  make  I  L  =  r  .  F  I.  Since 
the  triangles  I  H  K  and  I  L  C  are  similar,  we  have 

I  L  :  I  K  :  :  I  C  :  I  H 

whence  IL.IH  =  IK.ICorr.FI.IH  =  IK.R 

substituting  for  I K,  its  equal  L  M  and  dividing  by  r  we  have 

R 
F  I .  I  H  :-  -L  M  =  E  D  .  L  M. 

The  first  member  of  this  equation  is  equal  to  the  rectangle 
FG  H  I,  which  represents  the  rate  at  which  the  convex  sur- 
face is  generated,  and  the  last  member  is  composed  of  the 
constant  E  D  into  L  M  which  represents  the  rate  at  which 
the  diameter  is  generated.  Hence,  the  rates  of  generation  of 
the  convex  surface  of  the  ungula,  and  of  the  diameter  of  its 
base,  having  a  constant  ratio  from  the  beginning,  their  mag- 


DIRECT   METHOD    OF   RATES. 


29 


nitufles,  when  completed  will  (Axiom  II)  have  the  same  ratio;: 
therefore  the  convex  surface  of  the  ungiila  will  be  equal  to 
the  dicDneter  of  the  base  multiplied   by  its  extreme  height. 


Lemma  II. 

To  find   the   measure  of  the   volume  of  an  tmgula  tvith  a 

semicircular  base. 

Let  E  A  D  B  be  the  ungula,  of  which  the  semicircle 
A  D  B  is  the  base  and  E  D,  per- 
pendicular to  D  C  is  the  extreme 
altitude.  Let  F I  represent  one 
of  the  elements  of  the  convex 
surface,  and  join  F  and  I  to  C. 
Then  we  may  suppose  the  ungu- 
la to  be  generatefl  by  the  verti- 
cal right-angled  triangle  C  F  I 
revolving  about  the  center  C,  its 
base  being  one  of  the  radii  of  the  semicircle,  and  the  side  F  I 
being  always  vertical,  and  forming  one  of  the  elements  of 
the  convex  surface  of  the  ungula.  Draw  I  H  to  represent 
the  rate  of  generation  of  the  circumference  of  the  base  of 
the  ungula  ;  construct  the  rectangle  F  Gr  H  I  and  join  G  and 
H  to  C,  then  (Axiom  I)  the  triangle  I  H  C  will  represent  the 
rate  of  generation  of  the  base,  the  rectangle  F  G  H  I  will 
represent  the  rate  of  generation  of  the  convex  surface,  and 
the  pyramid  C  F  G  H  I  will  represent  the  rate  at  which  the 
volume  of  the  ungula  itself  will  be  generated  ;  for  while  the 
point  I  would  move  to  H  under  its  temlency  at  the  moment, 
the  line  F  I  would  move  to  G  H  describing  the  rectangle, 
and  the  triangle  C  F  I,  would  move  to  C  G  H  describing  the 
pyramid' C  F  G  H  I,  and  these  being  all  described  simultane- 
ously, at  a  uniform  rate,  will  represent  the  relative  rates  of 
generation  of  the  respective  magnitudes  to  which  they  be- 
long. 


30  DIRECT    METHOD    OF   EATES. 

Now  the  volume  of  the  pyramid  is  equal  to  its  base  F  G  H  I 
multiplied  by  one-third  of  its  altitude  I  C  or  R  ;  or,  in  other 
words,  the  rate  of  generation  of  the  volimie  of  the  ungula  is 
equal  to  that  of  the  convex  surface  multiplied  by  one-third 
of  the  radius  of  its  base,  which  is  therefore  the  constant 
ratio  between  these  two  rates,  and  hence  also,  (Axiom  II) 
the  ratio  between  the  completed  magnitudes.  So  that  the 
volume  of  the  ungula  is  equal  to  its  convex  surface  multi- 
plied by  one-third  of  the  radius  of  its  base — that  is  (Lemma  1), 
ED.AB.iR  =  ED.|R  ~— or,  its  extreme  altitude  multi- 
plied by  two-thirds  of  the  square  of  the  radius  of  its  base. 

Proposition^  YIII. 

To  find  the  area  of  the  surface  of  a  Sphere, 

Suppose  the  sphere  to  be  generated  by  the  revolution  of 
the  semicircle  C  B  D  (Fig.  8)  about  the  diameter  C  D.  Then 
the  semi-circumference  C  B  D  will  gene- 
rate the  surface  of  the  sphere.  Now 
every  point  in  the  curve  C  B  D  tends  to  a 
move  in  a  direction  perpendicular  to  its 
plane  at  a  rate  proportional  to  its  dis- 
tance from  C  D,  the  axis  of  revolution, 
and  under  this  tendency  it   would   in  fio.  8. 

in  some  unit  of  time  assume  the  position  of  the  semi-ellipse 
CE  D,  geuerating  at  the  same  time  the  convex  surface  of  the 
ungula  C  E  D  B,  which  is,  therefore,  the  symbol  (Axiom  I) 
representing  the  rate  of  generation  of  the  surface  of  the 
sphere,  while  the  line  E  B  described  by  the  point  B,  in  the 
middle  of  the  arc  C  B  D,  and  perpendicular  to  its  plane  is  the 
symbol  representing  the  corresponding  rate  of  generation  of 
the  circumference  of  a  great  circle.  But  the  convex  surface 
of  the  ungula  is  equal  to  its  extreme  height  E  B,  multiplied 
by  the  diameter  C  D,  which  is,  therefore,  the  constalit  ratio 
between  the  rates  of  generation  of  the  surface  of  the  sphere 


DIRECT   METHOD   OF   RATES.  81 

and  of  the  circumference  of  its  great  circle.  The  magni- 
tudes themselves  are,  therefore  (Axiom  IT),  in  the  same  r.itio, 
and  the  surface  of  a  sphere  is  equal  to  its  diameter  multiplied 
by  the  circumference  of  its  great  circle. 

Proposition"  IX. 

To  find  the  measure  of  the  volume  of  a  Sphere. 

The  sphere  being  supposed  to  be  generated  by  the  revolu- 
tion of  the  semicircle  C  B  D  about  the  diameter  C  D  (Fig.  8), 
when  it  is  revolving  towards  the  point  E,  every  point  in  it 
will  tend  to  move  in  a  direction  perpendicular  to  its  plane, 
and  at  a  rate  proportional  to  its  distance  from  C  D  the  axis  of 
revolution  ;  and  the  point  B  in  the  middle  of  the  arc  C  B  D 
will  tend  to  describe  the  line  B  E  perpendicular  to  the  plane 
of  C  B  D,  in  some  unit  of  time,  and  would  do  so  if  left  to  that 
tendency.  The  semicircle  C  B  D,  at  the  end  of  the  same 
unit  of  time,  would  be  found  in  the  ellipse  C  E  D,  having 
described  or  generated  the  ungula  E  C  B  D,  which  may,  there- 
fore (Axiom  1),  be  taken  as  the  symbol  of  the  rate  at  which 
the  volume  of  the  sphere  is  generated,  while  E  B  is  the 
symbol  of  the  rate  at  which  the  circumference  of  its  great 
circle  is  generated.  But  the  volume  of  the  ungula  is  equal 
to  its  extreme  height  E  B  multiplied  by  two-thirds  of  the 
square  of  the  radius,  which  is,  therefore,  the  ratio  between 
the  rates  of  generation  of  the  volume  of  the  sphere  and  of 
the  circumference  of  its  great  circle.  Hence  the  magnitudes 
themselves  are  in  the  same  ratio  (Axiom  II)  and  the  volume  of 
the  sphere  is  equal  to  the  circumference  of  its  great  cii'cle 
multiplied  by  two-thirds  of  the  square  of  its  radius — or  what 
is  the  same  thing,  to  its  surface  multiplied  by  one-third  of 
its  radius. 


32 


DIRECT   METHOD   OF   RATES. 


Fig.  9 


Proposition  X. 

To  find  the  area  of  an  Ellipse. 

Let  there  be  an  ellipse  (Fig.  9)  described  on  A  B  as 
its  transverse  axis  and  a  circle 
described  on  the  same  line  as  a 
diameter.  Suppose  the  ellipse  to 
be  generated  by  the  flowing  of 
the  double  ordinate  E  C  from  B  a  I 
towards  A,  and  the  circle,  by  its 
double  ordinate  D  H  flowing  in 
the  same  direction  ;  both  lines 
being  constantly  together  and 
remaining  perpendicular  to  the  axis  A  B.  Then  the  ratio  of 
the  rates  at  which  the  surfaces  are  generated  will  be  at  all 
times  the  same  as  that  of  their  respective  generatrices — that 
is,  as  D  L  :  EL:  :  R.O  :  M  0  :  hence  (Axiom  II)  the  surfaces 
generated  will  have  the  same  ratio.     But  the  surface  of  the 

circle  is  equal  to  tt  R  0  or  ^r  A  0  .  R  0  ;  hence  that  of  the 
ellipse  will  be  :r  A  0 .  M  0  ;  or,  tt  into  the  rectangle  of  the 
semi-axes. 


Proposition  XL 

To  find  the  measure  of  the  volume  of  an  Ellipsoid. 

Suppose  the  ellipsoid  to  be  generated  by  the  revolution  of 
a  semi-ellipse  A  M  B  (Fig.  9)  about  its  major  axis  A  B,  and 
suppose  a  sphere  A  R  B  H  to  be  described  about  the  same  axis 
as  a  diameter.  We  may  then  suppose  each  to  be  generated  by  a 
circle  flowing  with  the  same  uniform  velocity  from  B  toward 
A, — the  radii  D  L  and  E  L  of  the  circles  varying  so  that  the 
circumference  of  each  shall  pass  through  the  meridian  curve 
of  the  volume  generated  by  it.  Then,  since  the  circles  move 
with  equal  velocities,  the  ratio  of  the  rates  at  which  volumes 


DIPECT    METHOD   OF   RATES.  33 

are  generated  will  be  at  all  times  the  same  as  that   of  the 
generating  circles,  or  as 


DL:EL:  :R0  :M0 
and  hence  (Axiom   II)  the  ratio  of  the   volumes   themselves 
will  be  the  same. 

But  the  volume  of  the  sphere  is      ^ 


%  TT  R  0  or  'k  ^  A  0.  R  0 
and  therefore  the  volume  of  the  ellipsoid  is 

Vs  ^AO.MO' 
or  equal  to  |  of  that    of  a   cylinder    whose   altitude   is   the 
transverse  axis  and  whose  diameter  is  the  conjugate  axis  oi 
of  the  generating  ellipse. 


34  DIRECT   METHOD    OF    IJATES. 


THE 

DIRECT  METHOD  OF  RATES 

AS    APPLIED   TO 

SOME  PROBLEMS  IN  MECHANICS. 

To  find  the  Center  of  Gravity. 

Definition. — The  center  of  gravity  of  any  body  is  that 
point,  about  which,  if  supported,  the  body  will  be  evenly 
balanced,  so  that  it  will  have  no  tendency,  arising  from  its 
weight,  to  change  its  position. 

It  is  proven  in  mechanics,  that  if  the  weights  of  several 
bodies  be  multiplied  by  the  distances  of  their  respective  cen- 
ters of  gravity  from  a  given  vertical  plane,  the  sum  of  the 
products  will  be  equal  to  the  product  arising  from  multiply- 
ing the  sum  of  the  weights  of  the  bodies  by  the  distance  of 
their  common  center  of  gravity  from  the  same  plane.  Hence, 
if  we  divide  the  sum  of  these  products  by  the  sum  of  the 
weights,  we  obtain  the  distance  of  their  common  center  of 
gravity  from  that  plaue. 

The  product  of  the  weight  of  any  body  into  the  distance 
of  its  center  of  gravity  from  any  fixed  vertical  plane  is  called 
the  moment  of  that  body  with  respect  to  that  plane.  We 
say  therefore  that  the  sum  of  the  moments  of  all  the  bodies 
in  any  system  is  equal  to  the  moment  of  their  sum  ;  and  also 
that  the  sum  of  the  moments  of  the  several  parts  of  any 
single  body  is  equal  to  the  moment  of  the  body  itself. 

Tliis  principle  is  useful  in  determining  the  centers  of 
gravity  of  those  bodies  whose  dimensions  can  be  reduced  to 
geometrical  magnitudes.     Among  such  bodies  are  some  that 


DIRECI    METHOD   OF   RATES. 


35 


can  be  divided  into  parts  whose  respective  centers  of  gravity 
can  be  found  by  inspection,  and  thus  the  center  of  gravity  of 
the  whole  is  easily  obtained;  but  there  are  many  that  cannot 
be  thus  divided,  however  small  the  parts  may  be.  For  this 
latter  class  it  is  necessary  to  resort  to  the  ultimate  particles 
of  matter,  so  small  that  each  one  may  be  considered  a& 
occupying  the  same  place  as  its  center  of  gravity.  It  is,  of 
course,  impossible  to  estimate  the  weights  of  such  particles 
singl}',  or  the  sum  of  the  required  products,  by  any  direct 
process.  Hence  we  resort  to  one  that  is  indirect.  Tliat  is, 
we  find  the  rate  at  which  the  sum  of  these  products  increases^ 
and  from  that  rate  determine  the  sum  itself. 

In  the  following  examples,  it  is  assumed  tliat  the  lines  and 
surfaces  are  homogenous,  material  bodies,  of  uniform  thick- 
ness so  small  as  not  to  affect  the  distance  of  the  center  of 
gravity  from  the  plane  of  reference. 

To  find  the  center  of  gravity  of  a  curved  line. 

Let  B  A  C  be  the  curved  line  symmetrical  about  the  axis 
A  G ;  then  it  is  evident  that  its  center  of    c 
gravity  will  be  somewhere  in  that  axis,  and   ^ 
that  the  center  of  gravity  of  each  half  of  the 
curve  will  be  at  the  same  distance  from  the 
plane  of  reference,  D  E,  as  that  of  the  whole.    A 

We  will,  therefore,  take  the  part  A  B,  and 
suppose  it  to  be  generated  by  a  point  flowing 
from  A  toward  B  around  the  curve  at  a  rate 
determined  by  some  law.  From  B  draw  a 
line  B  0  tangent  to  the  curve  at  B.  Then  ^ 
the  generating  point  on  arriving  at  B  would  tend  to  flow  at 
a  rate,  and  in  a  direction,  which,  if  uniformly  continued, 
would  generate  a  tangent  line  as  B  0  in  a  unit  of  time;  and, 
therefore,  B  0  may  be  taken  to  represent  the  rate  of  increase 
of  the  curve  at  the  point  B:  —  that  is,  it  will  represent  the 
differential  of  the  curve.    Calling  the  weight  of  the  curve  Wy 


B 

^-^ 

f3 

\ 

\ 
\ 

C 

J 

36  DIRECT   METHOD    OF   HATES. 

this  would  be  represented  by  aw.  feince  the  line  B  0  repre- 
isents  a  rate  Avhich  is  something  which  exists  at  the  point  B 
only,  its  moment  will  be  expressed  by  B  0  X  B  m,  or  x  dw^ 
which  represents  therefore  the  moment  of  the  rate  of  genera- 
tion of  the  curve.  Now,  since  B  0  or  dw  represents  the  rate 
of  increase  of  the  curve,  it  will,  of  course,  represent  that  of 
the  sum  of  its  particles,  and  this  rate  multiplied  by  ^m  or  x 
{x  div)  will  represent,  not  the  sum  of  the  jjrodiids  of  the 
particles  into  their  respective  distances  from  D  E  which  we 
are  seeking,  but  their  rate  of  increase  or  the  differential  of 
that  sum:  so  that  the  moment  of  the  rate  of  increase  of  the 
<iurve  is,  also,  the  rate  of  increase  of  the  sum  of  the  moments 
of  its  particles,  and  its  integral  {/x  dw)  will  be  the  sum  of 
these  moments. 

If,  therefore,  we  divide  /  x  dw  by  the  mass  of  the  curve, 
we  shall  have  for  a  quotient  the  required  distance  of  its 
center  of  gravity  from  the  plane  of  reference. 

Note. — It  might  be  supposed  that  the  mnraeut  of  B  O  should  be  B  O 
{^  +  3^2  B  O)  since  the  center  of  gravity  of  B  O  is  in  the  middk'  of  its 
lengtli.  But  it  must  be  remembered  that  B  O  is  not  an  integral  part  of 
the  curve,  but  a  symbol,  or  an  ide((l  line  representing  a  supposition.  It  is 
a  measure  of  value;  but  it  is  a  value  existing  at  the  point  B,  and  there 
•onl}',  viz.:  that  of  the  rate  of  geneiatio  \  of  the  curve  at  the  moment  the 
generating  point  arrives  at  B.  Similarly,  if  a  falling  ])ody  acquire,  at  a 
point  A,  a  rate  of  motion  of  50  feet  in  one  second,  we  multipl}^  its  weight 
by  00  and  obtain  its  momentum.  Here  the  rate  of  motion,  represented 
by  50  feet,  and  the  corresponding  momentum  belong,  wholly,  to  the 
point  A  where  these  values  exist  at  the  moment. 

So  the  value  of  the  line  B  O  and  its  moment  belong,  wholly,  to  the 
point  B  where  the  rale  exists  whose  value  is  represenled  by  B  O. 

For  another  illustration:  suppose  a  body  to  be  genei'ated  in  a  dij-ec- 
tion  from  the  plane  of  reference  by  successive  accretions  of  one  unit 
each;  and  suppose  that  the  last  unit  had  been  geneiated  at  a  rate  that 
would  have  produced  n  such  units  in  one  unit  of  time,  then  n  units  will 
represent  the  rate  of  increase  of  the  last  unit,  which  is,  also,  that  of  the 
entire  body;  and  the  sum  of  the  products  of  all  the  units  into  the 
respective  distances  of  their  centers  of  gravity  from  the  plane  of  refer- 


DIRECT    METHOD    OF   RATES. 


37 


ence,  will  be  incrensing  at  the  same  rate  as  that  of  the  last  unit  (repre- 
sented by  n  such  units)  into  its  distance. 

But  this  is  nothing  more  than  the  moment  of  the  rate  of  increase  of  the 
sum  of  the  products,  which  is,  therefore,  equal  to  the  rate  of  increase  of 
the  sum  of  the  moments;  and  this  divided  by  the  sum  of  the  parts,  will 
give  the  rate  of  increase  of  the  distance  of  the  center  of  gravity  of  the 
whole  from  the  plane  of  reference. 

The  same  thing  may  be  expressed  more  briefly  as  follows:  As  the 
body  is  generated  and  eacii  successive  increment  is  multiplied  into  its 
distance  from  the  plane  of  reference,  the  rate  at  which  the  last  product 
is  formed  will  be  equal  to  the  rate  at  which  tiie  last  increment  is  formed 
multiplied  into  its  distance  from  the  plane  of  reference,  and  will  l)e  the 
rate  of  increase  of  the  sum  of  the  products.  Hence  the  moment  of  the 
rate  of  iiu  rease  of  the  body  is  equal  to  the  rate  of  increase  of  the  sum 
of  the  moments. 

To  apply  this  formula  we  obtain  the  value  of  dw  by  takins^ 
the  differential  of  the  length  of  the  line  from  its  equation, 
and  then  multiply  and  integrate  in  the  same  way  as  in  the 
infinitesimal  sj^stem. 


Proposition^. 
To  find  the  center  of  gravity  of  a  cycloid. 

Let  A  C  B  be  the  cycloid  of  which  C  F  is  the  axis.  A  B  is 
the  base  and  D  E  tan-        d  C  E 

gent  to  the  curve  at  the 
upper  extremity  of  the 
axis  and  parallel  to  the 
base.  As  the  cycloid  is 
symmetrical  about  its  A 
axis,  the  center  of  gravity  will  be  in  it,  and  ilie  problem  will 
be,  to  find  its  distance  from  C,  as  the  origin. 

This  distance  will  be  expressed  in  terms  of  tj,  but  as,  in  the 
equation  of  the  curve,  the  value  of  y  is  reckoned  from  the 
line  A  B,  we  must,  in  order  to  reckon  the  distance  from  C, 
^ubstitute  for  y  in  the  differential   equation  of  the  curve, 


38  DIRECT   METHOD    OF   RATES. 

another  y  or  y'  which  will  be  equal  to  2r  —  y;  whence  y  == 
2r-y''^. 

Mfiking  this  substitution  in  the  equation 


dx  =    y^y 


V2nj  -  if 
we  have 


"■i^. 


dx  =  d  (2>-  -  ./)  J^^^  =  -'^y\  ^^'       ^' 


y 


hence 

doL^  —  dy 
and 


,,  2r  -  ?/' 


But  we  have  (Art.  210  Cal.)  for  the  length  of  the  curve 


to  =  —9,  s/%'  (2r  -  y) 

reckoning   from    C;    and    substituting   2r   —   //'   for  y   we 
liave  w  —  —  2  \^%'y\  which  gives,  for  the  distance  of  the 

center  of  gravity  ironi  the  point  u,  '— ==  ' ^    ■'  - 

^  —  2v/2/-//' 

= ; — fy   ^2  dy    —  ^  f-^-  =  -3-  ij  ' 

—  2  v^//'  %    ^2 

Making  ?/'  =  2r  we  find  the  center  of  gravity  of  a  cycloid 
to  be  in  its  axis,  and  at  a  distance  from  its  upper  extremity, 
equal  to  one-third  of  the  diameter  of  the  generating  circle. 


DIEECT   METHOD   OF   RATES. 


;9 


To  find  the  cenfpr  of  gravity  of  a  plane  surface. 

Let  the  surface  be  bounded  by  the  curve  B  A  C  and  tlic 
double  ordinate  B  D  C,  and  be  symmetrical 
al)out  the  line  A  D.  This  line  will  then  con- 
tain the  center  of  gravity  of  the  surface,  and 
the  question  will  l)e,  to  determine  its  distance 
from  the  plane  of  reference  A  Y,  perpendicular 
to  A  D.  Since  the  center  of  gravity  of  each 
half  of  the  surface  is  at  the  same  distance  from 
the  plane  of  reference  as  that  of  the  whole,  we 
will  take  the  upper  half,  A  B  D,  for  our  pur- 
pose. Suppose  the  surface  to  have  been  gene- 
rated by  the  movement  of  the  ordinate  commencing  at  A, 
and  having  its  extremities  always  in  the  bounding  line  or 
curve  and  its  middle  point  in  the  axis  A  D,  until  it  arrives 
at  B  C.  Let  D  F  represent  the  rate  at  whicli  the  axis  A  D 
was  increasing  (or  dx)  when  the  generating  ordinate  arrived 
at  B  D  then  the  rectangle  D  B  E  F  will  represent  the  rate 
at  which  the  surface  was  generated  at  the  same  moment,  for 
it  is  the  surface  that  ivoidd  have  been  generated,  if  the 
ordinate  had  continued  to  move,  unchanged,  at  a  uniform 
rate  for  the  same  unit  of  time  as  that  required  by  D  F,  and 
this  rate  will  be  expressed  by  B  D,  D  F  or  ydx. 

Since  this  symbol  represents  that  which  exists  at  the 
ordinate  B  D,  its  moment  will  be  found  by  multiplying  it  by 
the  distance  of  B  D  from  the  plane  of  reference,  that  is, 
B  D,  D  F,  B  Y  or  xydx.  But  the  rectangle  represents  the 
rate  of  increase  of  the  sum  of  all  the  parts  of  the  surface, 
and  therefore,  the  product  of  the  rectangle  by  its  distance 
from  the  plane  of  reference  will  represent  the  rate  of  increase 
of  the  sum  of  the  products  of  all  these  parts  into  their 
respective  distances  from  the  same  plane;  that  is,  the  rate  of 
increase  (or  differential)  of  the  sum  of  the  moments.  Hence, 
fxydx  will  represent  the  sum  of  the  moments,  and  this 
divided  by  the  surface  will  give  the  distance  of  its  center  of 
gravity  from  the  plane  of  reference. 


40 


LIRECT    METHOD    OF   RATES. 


Example. 

To  find  the  center  of  gravity  of  a  stirface  hounded  hy  a 
parabola  and  its  double  ordinate. 

Let  the  plane  of  reference  be  tangent  to  the  parabola  at 
its  vertex.     In  this  case  xy  dx  becomes  ^^poi?U  dx  and  the 


surface  is  i  xy  —  i  ^2p  x '/«  and  hence  for 


fxydx 


we  shall 


have 


f'^^p  x^U  dx  __  fx^U  dx   _  Vs  x  Va  _ 
iV^px^h     -  -  - 


Jl.  /yi  3  / 

3  U/    /j 


^xV. 


%  X  +  C. 


But  when  a:;  =  0  we  have  s  =  0  and  hence  C  =  0  and  the 
<iistance  of  the  center  of  gravity  of  a  segment  of  a  parabola 
from  its  vertex  is  %  the  length  of  its  axis. 

To  find  the  center  of  gravity  of  a  surface  of  revolution. 

Suppose  the  surface  to  be  generated  by  the  flowing  of  the 
circumference  of  a  circle,  constantly  parallel  to  itself,  and 
constantly  intersecting  the  meridian  section  of  the  surface, 
and  whose  center  shall  be  always  in  the  axis  of  revolution. 
Let  C  A  F  represent  the  meridian  section  of  the  surface; 
A  B,  its  axis  of  revolution;  A  Y,  the  plane  of  reference; 
C  F,  the  diameter  of  the  generating  circum- 
ference; 0,  the  length  of  the  meridian  curve: 
and  S,  the  area  of  the  surface. 

Suppose  the  center  of  the  generating  circum- 
ference to  have  arrived  at  the  point  B  in  its 
movement  awa}''  from  the  plane  of  reference. 
At  that  instant  every  point  in  this  circumfer- 
ence will  have  generated  a  meridian  curve,  and 
will  tend  to  move  in  the  direction  of  the  tan- 
gent to  the  one  in  which  it  lies,  and  at  a  rate 
equal  to  that  at  which  the  curve  was  generated.     Thus  the 


Y 

C 

D 

A 

/" 

E 
B 

F 

DIRECT   METHOD    OF   RATES.  41 

point  C  will  tend  to  move  in  the  direction  C  D,  tanp:ent  io 
the  curve  at  C.  Let  C  D  represent  the  rate  at  which  the 
point  C  is  moving  when  it  has  arrived  at  that  position;  then 
the  rate  of  movement  of  every  point  in  the  generating 
circumference  will  be  the  same;  and  the  rate  of  generation 
of  the  surface  will  be  represented  by  the  surface  of  a  cylinder 
whose  base  is  the  generating  circumference,  and  whose 
height  will  be  equal  to  C  D  (or  the  differential  of  0)  which 
represents  the  rate  at  which  every  one  of  its  points  moves. 
This  figure  will  be  a  cylinder  because  the  surface  which 
represents  the  rate,  at  that  moment,  must  be  that  which 
would  be  generated  at  a  uniform  equal  rate,  which  requires 
that  the  generating  circumference  should  not  change  its 
dimensions.  Hence  if  we  make  C  E,  parallel  to  the  axis,  = 
C  D,  the  rate  of  generation  of  the  surface  of  revolution  at 
the  moment  the  center  of  the  generating  circumference 
arrives  at  B  will  be  represented  by  the  surface  of  a  cylinder 
of  w^iich  C  F  is  the  diameter  of  the,  base  and  C  E  the 
altitude. 

Now  the  surface  of  this  cylinder  being  the  symbol  of  the 
rate  of  increase  of  the  sum  of  all  the  particles  of  the  surface, 
will  be  represented  by  ds;  and  if  we  multiply  it  by  the  dis- 
tance of  C  from  the  plane  of  reference,  the  product  {xds)  will 
be  the  symbol  of  the  rate  of  increase  of  the  sum  of  all  the 
products  arising  from  multiplying  each  particle  into  its 
distance  from  the  plane  of  reference  —  that  is,  the  moment 
of  the  rate  of  increase  of  the  sum  of  the  particles  is  the  rate 
of  increase  of  the  sum  of  the  moments  of  the  particles, 
which  will  be  represented  by 


xds  =  X.  2-^1/  ^dx^  +  ^^y'  =  ^^xy  [dot?  +  dif  )  Vj 

and  the  integral  of  this  divided  by  the  surface  will  give  the 
distance  of  its  center  of  gravity  from  the  plane  of  reference. 


42 


DIRECT   METHOD   OF   RATES. 


Example. 
To  find  the  center  of  gravity  of  the  surface  of  a  spherical 

segment. 

Let  A  B  be  the  axis  of  the  segment  perpendicular  to  its 
base  and  A  Y  the  plane  of  reference;  when  the 
center   of  gravity   will   be  in   A  B,  and  since 
y2  _  ^.2  _  ^2  ^Y\Q  differential  of  the  curve  {dO) 
will  be  equal  to 


4 


x^  dx^ 


+  dx^ 


-    ^^>^^.2_, 


'r"  —  x^     '  Xi  /•"  —  x" 

and  the  differential  of  the  surface  (ds)  will  be 


^-■y^-^ 


—  x" 


.  dx  =  27crdx 


while  the  surface  itself  is  equal  to 

2  7:  r  X 

hence  for  the  distance  of  the  center  of  gravity  from  the 

/  /  X  d  s\ 
plane  of  reference   I- — ; — I   we  have 

f2-^rxdx  _   fxdx  _  x 

.  _ 


''l-Krx 


X 


or  the  center  of  gravity  of  the  surface  of  a  spherical  segment 
is  at  a  distance  from  its  vertex  equal  to  half  its  height;  and 
making  :r  =  R,  the  center  of  gravity  of  the  surface  of  a 
hemisphere  is  on  the  middle  of  the  radius  perpendicular  to 
its  base. 


DIRECT    METHOD    OF   RATES. 


To  find  the  center  of  gravity  of  a  solid  of  revolution. 

Let  CAB  represent  the  meridian  curve  of  the  solirl,  A  F 
its  axis.  A  Y  the  phme  of  reference,  Gr  the 
center  of  gravity  in  the  axis  A  B.  Call  the 
solid  V,  and  suppose  it  to  he  generated  by  a 
circle  flowing  constantly  parallel  to  itself  from 
the  vertex  A,  and  varying  its  diameter  in  such 
a  way  that  its  circumference  shall  always  inter- 
sect the  meridian  curve  on  the  surface  of  the 
solid.  Then  if  D  F  represent  the  rate  of  move- 
ment of  the  center  of  the  circle  (or  dx)  when  it 
is  at  D,  the  cylinder  having  B  D  for  the  radius 
of  its  base  and  D  F  for  its  altitude,  will  represent  the  rate  of 
generation  of  the  solid  at  the  same  moment,  and  will  be 
equal  to  Ttij^dx.  Multiplying  this  by  x,  we  have  the  moment 
of  the  rate  of  increase,  which  as  we  have  seen,  is  the  rate  of 
increase  of  the  sum  of  the  moments  of  all  the  particles  in 
the  solid;  and  the  integral  of  this  is  the  sum  of  the  same 
moments,  and  equal  to  the  moment  of  the  entire  body,  or 

V  X  AG;  hence 

^^  _  fiii/xdx 


Example. 

To  find  the  center  of  gravity  of  a  hemisphere  with  the  origin 
at  the  center  and  the  plane  of  referefice  i)assing 
through  the  origin. 
The  volume  of  the  spherical  segment  next  to  the  center  is 

and  we  have  i'or  f-^y^xdx 

f^  (K'  -  x' )  xdx  =y'-  (R2.r  -  x')  dx  =  ^  fe'^  -  '^j 


4:4  DIRECT   METHOD    OF   RATES. 

whence 

2         4 

j^yj    z=     -^ —  


.  (r^.  -  5) 


whicli  becomes,  by  making  x  =  R 


3 


AG  -  p^3  _  ,^^^3  -  gR. 

or  the  center  of  gravity  of  a  hemisphere  is  on  the  radius 
perpendicular  to  its  base  and  three-eighths  of  its  leugth  from 
the  center  of  the  sphere. 


CENTER   OF    OSCILLATION". 

A  simple  pendulum  is  a  particle  of  matter  suspended  by 
an  imponderable  rod  to  a  point  about  which  it  can  vibrate 
freely. 

A  compound  pendulum  is  any  mass  of  matter  suspended  on 
a  horizontal  axis  which  does  not  pass  through  its  center  of 
gravity,  and  about  which  it  can  vibrate  freely. 

The  center  of  oscillation  of  a  pendulum  is  that  point  in 
which,  if  the  entire  mass  be  concentrated,  the  pendulum 
would  vibrate  in  the  same  time  as  it  would  in  its  natural 
state.  Hence  the  distance  of  the  center  of  oscillation  from 
the  axis  of  suspension  is  the  length  of  a  simple  pendulum 
vibrating  in  an  equal  time;  and  since  the  compound  pendu- 
lum vibrates  in  the  same  time  as  the  simple  one,  the  force  of 
gravity  must  act  on  each  at  the  same  distance  from  the  axis 
of  suspension;  that  is  the  resultant  of  all  the  forces  of  grav- 
ity, acting  upon  the  particles  of  a  compound  pendulum,  must 
pass  through  a  point  at  the  same  distance  from  the  axis  of  sus- 
pension, as  the  particle  which  composes  a  simple  penduhuu 
vibrating  in  the  same  time.     Hence  to  find  the  center  of 


DIRECT   METHOD   OF   RATES. 


45 


oscillation  we  must  find  through  what  point  this  resultant 
passes. 

Let  S  A  be  a  pendulum  composed  of  simple 
particles  arranged  in  a  straight  line  and  sus- 
pended at  the  point  S.  Let  injy  represent  the 
direct  action  of  the  force  of  gravity  or  weight 
of  the  particle  j;;  it  may  be  resolved  into  two 
component  parts,  viz.  np,  acting  in  the  direction 
of  the  length  of  the  pendulum,  and  j^^i  acting 
in  a  direction  perpendicular  to  its  length  and 
which  causes  it  to  vibrate.  This  component, 
pk  of  the  force  of  gravity  Avill  be  the  same  for 
every  particle  in  the  pendulum,  and  would  cause  it  to  move 
so  as  to  be  constantly  parallel  to  itself;  but  its  action  will  be 
resisted  by  the  axis  of  suspension,  and  thus  cause  the  pendu- 
lum to  turn  around  that  axis;  and  the  difference  between  the 
component  of  gravity  pk  and  the  resistance  at  S  will  be  for 
each  particle  the  effective  force  arising  from  its  weight  which 
tends  to  produce  the  vibration,  and  which  increases  with  its 
distance  from  S.  Hence  the  effective  force  of  any  particle, 
as  p  to  cause  a  vibration  may  be  represented  by  the  weight  of 
that  particle  into  its  distance  from  the  point  of  suspension. 
This  is  called  the  moment  of  force  of  the  particle  with  refer- 
ence to  the  axis  at  S.  Since,  at  every  instant,  these  moments 
of  force  are  acting  in  a  direction  perpendicular  to  the  line  of 
the  pendulum,  they  will  be  parallel  to  each  other,  and  the 
distance  from  their  resultant  to  S  will  be  found  by  multiply- 
ing every  one  into  its  distance  from  S,  and  dividing  the  sum 
of  the  products  by  the  sum  of  the  moments.  The  product 
of  a  moment  of  force  into  its  distance  from  S  is  called,  the 
moment  of  inertia  of  the  particle  with  reference  to  the  axis 
S,  which  is  therefore  equal  to  the  weight  of  the  particle  into 
the  square  of  its  distance  from  the  axis,  and  if  o  represent 
the  center  of  oscillation  we  shall  have 


46  DIRECT   METHOD   OF   RATES. 

sura  of  moments  of  inertia 


So 


sum   of  moments  of  force' 


wliicli  is  equal  to  the  sum  of  the  products  arising  from 
multiplying  each  particle  of  matter  into  the  square  of  its 
distance  from  the  axis  of  suspension  and  dividing  the  sum 
of  the  products  by  the  mass  multiplied  by  the  distance  of  the 
center  of  gravity  from  the  same  point. 

It  is  of  course  impossible  to  obtain  the  sum  of  these  pro- 
ducts directly,  since  the  particles  of  matter,  although  real 
are  too  minute  to  be  estimated  singly.  We  must  therefore 
proceed  indirectly  by  obtaining  the  rate  at  which  these  pro- 
ducts increase,  compared  with  that  at  which  the  distance  of 
the  center  of  oscillation  increases,  supposing  the  pendulum 
to  be  in  a  state  of  growth  or  increase. 

Let  S  A  be  a  straight  inflexible  pendulum  composed  of  a 
single  line  of  particles  of  matter,  and  suspended  at  the 
point  S.  Since  the  weight  of  any  part  of  this  pendulum 
will  be  in  proportion  to  its  length,  and  vice  versa ^  the 
one  may  be  represented  by  the  other,  and  the  center  of 
gravity  will  be  at  the  middle  point.  Let  that  point  be 
€,  and  let  0  be  the  center  of  oscillation;  while  the 
weight  of  the  pendulum  is  represented  by  iv.  Suppose 
now,  the  pendulum  to  be  generated  by  growing  from 
the  point  S  downwards  until  it  has  reached  the 
extremity  A;  and  when  at  that  point,  let  A  B  represent  °  ^ 
the  part  that  would  be  generated  in  a  unit  of  time  at  the 
rate  then  existing  uniformly  continued;  then  A  B  will  repre- 
sent the  rate  of  increase  of  the  pendulum  at  the  point  A  — 
that  is  to  say,  the  difierential  of  to;  and  since  the  weight  and 

length  are  proportional,  A  B  multiplied  by  S  A  (or  tv^dtv) 
will  represent  the  rate  of  increase  of  the  sum  of  the  products 
arising  from  multiplying  each  particle  into  the  square  of  its 
distance  from  the  axis  of  suspension;  and  the  integral  of 
this,  or  fiv^dw  will  be  the  sum  itself.     Dividing  this  sum  by 


DIRECT   METHOD    OF   RATES.  47 

the  weight  of  the  entire  mass  into  the  distance  of  its  center 
of  gravity  from  the  axis  of  suspension  —  that  is,  by  ?r  Vs  iv^ 
or  ^aw^  we  have  the  distance  of  its  center  of  oscillation 

from  the  axis;  that  is,  S  0  —   -,-; — 9-   =  77  w.     Hence  the 

center  of  oscillation  of  a  straight  pendulum  of  uniform 
weight,  and  without  thickness,  is  at  a  distance  from  its  point 
of  suspension,  equal  to  two-thirds  of  its  length. 


48  FORCE,  TIME  AND  VELOCITY. 


FORCE,  TIME  AND  VELOCITY. 

Force  is  that  which  tends  to  produce  or  destroy  motion. 
Momentum  is  sometimes  considered  as  a  force,  since  when  it 
exists  in  one  body,  it  can  impart  motion  to  another  by 
impact.  But  in  such  a  case  there  is  no  motion  produced  — 
that  is  to  say  —  the  quantity  of  motion  is  no  greater  after 
impact  than  before;  it  is  merely  divided  into  parts:  hence 
momentum  is  not  properly  a  producer  of  motion,  and  there- 
fore is  not  ?i  force. 

The  quantity  of  motion  or  momentum  of  a  body  is  a  com- 
plex idea,  of  which  the  elements  are,  its  mass  or  quantity  of 
matter,  and  its  velocity  or  rate  of  motion.  For  a  given  mass 
the  momentum  is  in  proportion  to  the  velocity,  and  for  a 
given  velocity  it  is  as  the  mass:  hence  the  multiplying  of 
either,  multiplies  the  momentum;  and  if  they  both  vary,  we 
must,  to  express  the  idea,  represent  the  momentum  in  the 
form  of  the  product  of  two  factors.  There  cannot  of  course 
be  a  real  product  of  two  such  quantities,  considered  abstractly, 
any  more  than  we  can  multiply  an  hour  by  a  pound;  nor 
can  any  siunle  product  represent  an  abstract  momentum;  we 
can  only  put  them  into  W\q  form  of  a  product  retaining  both 
the  factors.  So  if  m  represent  the  mass  and  v  the  velocity, 
the  momentum  will  be  represented  by  mv.  Uniform  velocity 
is,  also,  a  complex  idea,  being  the  relation  between  the  space 
passed  over  at  a  uniform  rate  by  a  moving  body,  and  the 
time  occupied  by  its  passage.  For  a  given  time  the  velocity 
will  be  directly  as  the  space,  and  for  a  given  space  the 
velocity  will  be  inversely  as  the  time.  Hence  the  relation 
between  the  two  will  be  represented  in  the  form  of  a  fraction 

S 
thus,  7p  in  which  the  numerator  represents  the  spp-oe  and  the 


FORCE,    TIME   AND   VELOCITY.  49 

denominator  the  time.    Here,  again,  it  must  not  be  supposed 

that  there  can  be  a  real  division  of  the  space  by  the  time; 

the  quantities  being  wholly  different  in  kind,  one  can  no 

more  be  divided  by  the  other  than  a  mile  can  be  divided  by  a 

pound.     Nor,  indeed,  can  the  ahsolnte  value  of  the  velocity 

be  expressed  by  a  single  quotient  if  it  could  be  found,  since 

both  quantities  must  be  expressed  or  understood  in  order  to 

express  the  relation  between  them. 

To  give  an  exact  and  simple  measure  of  velocity,  we  use 

some  unit  of  time,  such  as  one  second,  one  hour,  one  day  or 

some   other   definite   period,   and    the  space  indicating   the 

velocity  will  be  that  which  is  passed  over  during  that  unit 

of  time;  but  the  space  must  be  associated  with  the  time  or 

S 
it  would  be  meaningless.   Hence,  in  the  expression  nn,  where 

S  represents  any  space  whatever  and  T  any  time  "whatever, 
we  divide  them  both  by  the  number  of  units  in  the  time, 
which  will  give  the  space  passed  over  in  one  unit  of  time 
and  thus  express  the  velocity  by  a  single  term.  If,  for 
instance,  the  space  is  five  hundred  feet  and  the  time  twenty 
seconds,  we  divide  by  twenty  and  the  result  will  be  twenty- 
five  feet  in  the  numerator  and  one  second  in  the  denominator, 
and  the  value  of  the  velocity  thus  expressed  is  twenty-five 
feet  in  one  second. 

Now  the  differential  of  any  quantity  is  represented  b}'  the 
change  that  would  take  place  in  a  unit  of  time  at  the  exist- 
ing rate  uniformly  continued  for  that  time;  hence  in  this 
case  the  numerator  of  the  fraction  is  the  differential  of  the 
space,  for  it  is  the  space  that  would  be  uniformly  passed  over 
in  one  unit  of  time,  and  the  denominator  is  the  difierential 
of  the  time  since  it  is  the  time  that  elapses  during  that  same 

unit,  and  therefore  the  velocity  will  be  expressed  by  -Tm. 

Since  the  rate  at  which  time  passes  never  varies,  d  T  will  be 
a  constant  quantity,  and  if  the  velocity  is  constant  then  d  S 


50  FOECE,    TIME   AND   VELOCITY. 

d  S 
will  also  be  constant,  and  the  expression  y^  may  be  replaced 

S 
by  m  for  the  increments  of  the  space  and  time  will  always 

be  proportional  to  their  rates  of  increase.    But  if  the  velocity 

is  variable  then  it  can  be  expressed,  for  any  instant,  only  by 

(i  S  . 

-Tm  in  which  case  d  S  does  not  indicate  the  space  actually 

passed  over  in  one  unit  of  time,  but  the  space  that  ivould  he 
passed  over  at  the  existing  rate  continued  for  that  unit.  It 
will  therefore  itself  be  a  variable;  and  the  velocity  at  any 
moment  therefore,  instead  of  being  represented  by  the  space 
divided  by  the  time,  will  be  represented  by  the  rate  of  change 
in  the  space  divided  by  the  rate  of  change  in  the  time. 

A  constant  force  free  to  act  will  always  be  measured  by  its 
effect;  that  is,  by  the  momentum  produced  in  a  unit  of  time, 
and  if  it  continues  to  act  it  will  produce  an  equal  amount  of 
momentum  in  each  successive  unit  of  time;  and  hence  at  the 
end  of  T  units  of  time  the  entire  momentum  will  be  repre- 
sented by  the  force  and  time  combined  in  the  form  of  a  pro- 
duct; as/*T.  This  method  of  representing  momentum,  like 
the  former,  is  to  show  how  the  two  ideas  of  force  and  time 
are  combined,  and  does  not  mean  that  there  can  be  a  real 
product  any  more  than  before:  the  momentum  is  still  a  com- 
plex idea,  but  the  elements  are  different  from  the  former 
ones. 

The  velocity  produced  by  a  continuous  force  will  of  course 

be  a  variable  one,  but  at  any  one  instant  the  momentum  will 

be  represented  by  mv^  v  being  the  velocity  at  that  instant, 

and  as  we  have  just  seen  will  also  be  represented  by  /T, — 

/T  i)iv 

we  have  mv  —  fl^  or  v  —  '■ —  or  /  =  -yfr,  in  which  the 

numerator  represents  the  momentum  produced,  and  the 
denominator  the  time  occupied  in  producing  it.  If  we  divide 
both  the  numerator  and  the  denominator  by  the  number  of 


FORCE,    TIME    AND   VELOCITY.  51 

units  in  the  time,  we  shall  have  the  momentum  produced  in 
one  unit  of  time  for  the  numerator,  and  the  unit  of  time 
itself  for  the  denominator;  and  these  will  represent  the  rate 

of  change  in  each,  and  hence  /  =  —pK  (^^^^  force  being  a 

uniform  one,  and  m  being  constant)  will  be  the  same  as  /  = 

-j^,  since  the  increments  being  made  at  a  uniform  rate  are 

always  proportional  to  the  rates  themselves. 

If  however  the  force  is  variable  its  value  can  be  expressed 

'})i  civ 
only  by     .  ^n    and  that  but  at  one  point  of  time,  for  7n  dv 

would  not  indicate  the  momentum  actually  acquired  in  one 

anit  of  time,  but  that  which  ivould  he  acquired  if  the  force 

were  to  continue  uniform  for  that  unit  of  time,  from  the 

instant  at  which  the  value  of  the  force  is  required.     If  then 

we  represent  a  variable  force  by  F  we  shall  always  have  F  — 

m  dv         .         ,         r/  S        ,         ^^       md^  S 

■  .rn   or,  Since  dv  =  -7777  we  have  r  =  — rm?. 

d  T  d  1  d  T^ 

If  the  force  is  that  of  gravity  then  the  mass  (or  m)  may 

be  omitted,  as  it  is  always  proportional  to  the   force,  and 

since  that  force  is  practically  constant  near  the  surface  of 

V 

the  earth,  we  have  in  that  case  (calling  the  force  g)  y  —  rf\ 

or  V  =  gH.    The  force  of  gravity  being  variable  at  a  distance 

from  the  earth,  if  we  represent  it  by  G  we  shall  have  G  = 

c^'^S         dv  ,    .  r/S        ,         G        dr.dT        dv 

j^,  =  ^;  andsmcei;  =  j^  we  have  -  =  ^^^^.  ^  ^ 

hence  Gf/S  —  vdv  and  G  =  ~Tq'- 

To  find  the  time^  space  and  velocity  near  the  earth^s  surface. 

Since  v  =  yT,  and  also  v  =    j^  we  have  g  T  =:  -j^  or 

rf  S  —  ^  T  (/  T  which  becomes  by  integration  S  =  ^  -^. 


52  FOKCE,    TIME   AND    VELOCITY. 

From  these  equations  we  can  determine  any  two  of  these 
quantities,  having  the  third  given. 

To  find  the  velocity  of  a  body  falling  through  a  great  distance. 

Let  r  be  the  radius  of  the  earth,  g  the  force  of  gravity  at 

its  surface,  G  the  force  at  a  distance  x  from  the  center  of  the 

earth,  where  the   body  is   supposed   to   be   after  falling   t 

seconds,  and  a  its  distance  when  the  fall  commences.     Then 

since  the  force  of  gravity  varies  inversely  as  the  square  of 

the  distance  from  the  center  of  the  earth  we  shall  have 

11  a  T^ 

O  :  (7  :  :  — 5  :  — 9,  wherefore  Gr  =  '■^— o-.     But  we  have  found 

G  =  -T^  and  equating  these  values  of  G  we  have 

vdv  _  gr'^ 

d^~'^' 

Since  S  =  a  —  a;  we  have  fi?  S  =  —  dx  and  by  substitution 

vdv  =  —       ..      =  —  gy^  x  ^  dx 
x^  ^ 

which  becomes  by  integration 

----  =  gr'^  x~^  or  v^  =  — h  C. 

2        "^  X 

2gr^ 
when  ?;  —  0  we  have  C  —  —  — —  and  therefore  the  velocity 

a  "^ 

acquired  by  a  body  falling  from  rest  will  be 

2  _  2£r_^  _  ^gr  ''^  _  '-Igr  '^  [a  —  x) 
X  a  ax         * 

If  iT  =  r  and  the  body  has  arrived  at  the  surface  of  the 
earth,  we  have 

2(7>-  (a  —  r) 
V  =       ~ — ^^ -. 


FORCE,    TIME    AND    VELOCITY.  63 

If  the  body  should  fall  from  the  moon,  we  should  have 
r  =  3956  miles,  a  =  60/-  and  g  =  32V«  feet  (that  being  the 
velocity  produced  by  the  force  of  gravity  in  one  second  of 
time);  and  for  the  value  of  v 


-J 


27.r59/-  |59>-.32Ve.A        .  q^      .,  , 

-  =  D.88  miles  nearly 


60/-  \        30 


r  being  taken  3956  miles. 

If  we  consider  the  distance  a  as  infinite,  it  does  not  follow 
that  the  velocity  of  a  body  falling  to  the  earth  through  an 
infinite  distance  will  be  infinite;  for  then  we  shall  have 


,2   _ 


=  2gr  and  v  =  v^2r//"  =  6.97  miles 

nearly;  so  that  however  far  a  body  may  fall,  on  arriving  at 
the  surface  of  the  earth,  it  can  never  attain  a  velocity 
greater  than  6.97  miles  in  one  second  of  time. 

To  find  the  force  of  attraction  excited  by  a  line  upon  a  pomt 
witJwut  it^  in  a  direction  perpendicular  to  the  line. 

Let  A  B  be  the  line  and  P  the  point  attracted  by  it.  Let 
P  A  be  perpendicular  to  A  B  and 
equal  to  a.  Suppose  the  line  to  be 
generated  by  a  point  flowing  from 
A  toward  B,  and  suppose  that  on 
arriving  at  B  it  is  moving  at  a  rate 
,that  would  carry  it  to  C  in  a  unit 
of  time;  then  B  C  will  represent  the  rate  of  increase  of  the 
line,  or  dx.  We  must  now  enquire  what  will  be  the  corre- 
sponding rate  of  increase  of  the  attraction  of  the  line  for 
the  point  P  in  the  required  direction.  The  attraction  of  B  C 
for  the  point  P  will  be  proportional  to  the  mass  of  B  C 

2 

divided  by  P  B  ;  for  B  C  is  not  an  integral  part  of  the  line 
A  B,  but  a  symbol  representing  a  value  which  exists  at  the 
point  B  only. 


54:  .  FORCE,   TIME   AND   VELOCITY. 

Let  m  represent  the  mass  of  one  unit  of  the  line  B  C  sup- 
posed to  exist  at  B,  and  m'  that  of  P;  and  let  k  represent 
the  attraction  of  one  particle  of  each  uiass  on  one  particle 
of  the  other;  then  the  attraction  of  one  unit  of  B  C  for  P 

will  be  equal  to         o  :  and  since  the  rate  of  increase  of  this 

PB 
attraction  is  proportional  to  the  rate  of  increase  of  the  line 
itself,  the  attraction  of  the  symbol  B  C  (which  represents 

this  rate,  or  dx)  for  the  point  P  will  be  equal  to  —       ..    . 

PB 
Now  this  expresses  the  attraction  of  B  C  for  P  in  the  direc- 
tion P  B,  and  is  to  its  attraction  in  the  direction  P  A  as  P  B  : 
PA::  ^a^  +  x^  :  a.  Hence  the  attraction  of  B  C  for  P  in 
the  direction  P  A  is  equal  to  its  attraction  toward  B  multi- 
plied by  a  and  divided  by  N^tr  +  x"^ . 

But  the  attraction  of  B  C  is  the  rate  of  increase  of  the 

,.,,,.  ,  .  ,  ,     kmm' dx  a 

attraction  oi  the  Ime,  and  is  equal  to  ■  2    . — 2   ^ 


kmm' adx       ^  -  ^    -,                1      •    .         ,.             kmm' x 
~  /  2  _i      2  \  8/  which  becomes,  by  integration, r 

[a    -\-  X  )  li  a  ^(r  -\-  x? 

(C  being  zero)  which  is  the  whole  attraction  of  A  B  for  P  in 
the  direction  P  A  supposing  re  =  A  B. 


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